simple steps to success in algebra

Copyright© 2009 Algebra-class.com

Simple Steps to Success in Algebra

By Karin Hutchinson


Table of Contents

Algebra Readiness Test .................................................................................................................................................................... 4

Basic Algebra ................................................................................................................................................................................. 11

Integers and Absolute Value ............................................................................................................................................... 11

Adding Integers .................................................................................................................................................................. 13

Subtracting Integers ............................................................................................................................................................ 15

Multiplying and Dividing Integers…………………………………………………………………………………………… 16

The Distributive Property………………………………………………………………………………………………………17 Simplifying Algebraic Expressions…………………………………………………………………………………………… 20 Solving Equations……………………………………………………………………………………………………………………....24 Balancing Equations……………………………………………………………………………………………………………24 One-Step Equations…………………………………………………………………………………………………………….26 Two Step Equations…………………………………………………………………………………………………………… 38 Equations with Fractions……………………………………………………………………………………………………….46 Literal Equations…………………………………………………………………………………………………………….…52 Equations with Variables on Both Sides……………………………………………………………………………………….55 Word Problems…………………………………………………………………………………………………………………59 Graphing Equations……………………………………………………………………………………………………………………66 Table of Values…………………………………………………………………………………………………………………66 Calculating Slope……………………………………………………………………………………………………………….72 Using Slope Intercept Form…………………………………………………………………………………………………….76 Rate of Change…………………………………………………………………………………………………………………82 Standard Form……………………………………………………………………………………………………………….…89 Using X and Y Intercepts………………………………………………………………………………………………………90 Writing Equations………………………………………………………………………………………………………………………98 Slope Intercept Form……………………………………………………………………………………………………….…..98 Standard Form………………………………………………………………………………………………………………...104 Word Problems………………………………………………………………………………………………………………..107 Using Slope and a Point………………………………………………………………………………………………………109 Given Two Points…………………………………………………………………………………………………………..…114 Systems of Equations……………………………………………………………………………………………………………….....118 Graphing Systems of Equations………………………………………………………………………………………………118 Using Substitution Method……………………………………………………………………………………………………125 Using Linear Combinations…………………………………………………………………………………………………...130 Real World Problems…………………………………………………………………………………………………………136 Inequalities…………………………………………………………………………………………………………………………….141 Inequalities in One Variable………………………………………………………………………………………………..…141 One Variable Word Problems……………………………………………………………………………………………..….150 Graphing Inequalities…………………………………………………………………………………………………………156 Graphing Systems of Inequalities…………………………………………………………………………………………..…164 Systems of Inequalities Word Problems………………………………………………………………………………………170 Functions……………………………………………………………………………………………………………………………….174 Relations and Functions………………………………………………………………………………………………………174 Vertical Line Test…………………………………………………………………………………………………………..…176 Function Notation…………………………………………………………………………………………………………..…180 Linear Functions………………………………………………………………………………………………………………183 Linear Functions Part 2……………………………………………………………………………………………………….184 Quadratic Functions………………………………………………………………………………………………………..….190 Vertex Formula………………………………………………………………………………………………………………..195 Step Functions………………………………………………………………………………………………………………...201


Exponents & Monomials …………………………………………………………………………………………………………….205 Exponents…………………………………………………………………………………………………………………….205 Laws of Exponents……………………………………………………………………………………………………………208 What is a Monomial…………………………………………………………………………………………………………..212 Multiplying Monomials………………………………………………………………………………………………………214 Dividing Monomials………………………………………………………………………………………………………….217 Simplifying Monomials………………………………………………………………………………………………………220 Negative Exponents & Zero Exponents………………………………………………………………………………….…...223 Scientific Notation……………………………………………………………………………………………………………227 Scientific Notation & Multiplying and Dividing…………………………………………………………..…………………236 Polynomials……………………………………………………………………………………………………………………………242 Intro (Definitions)…………………………………………………………………………………………………………….242 Adding Polynomials………………………………………………………………………………………………………….244 Subtracting Polynomials………………………………………………………………………………………..……………..247 Multiplying Polynomials………………………………………………………………………………………….……….….251 FOIL Method……………………………………………………………………………………………………………….…253 Squaring a Binomial – A Special Case……………………………………………………………………………………..…256 Difference of Two Squares – A Special Case……...…………………………………………………………………………259 Factoring Polynomials………………………..……………………………………………………………………………….262 Factoring by Grouping….…………………………………………………………………………………………………….268 Factoring Trinomials………………………………………………………………………………………………………….271 Quadratic Equations…………………………………………………………………………………………………………………..276 Intro (Definitions)……………………………………………………………………………………………………………..276 Introduction to Square Roots………………………………………………………………………………………………….277 Solving Simple Quadratic Equations………………………………………………………………………………………….280 Pythagorean Theorem…………………………………………………………………………………………...…………….283 Factoring Quadratic Equations………………………………………………………………………………………………..286 Quadratic Formula…………………………………………………………………………………………………………….289 Graphing Quadratic Equations………………………………………………………………………………………………..293 Probability…………………………………………………………………………………………………………………………..…301 Simple Events…………………………………………………………………………………………………………………302 Fundamental Counting Principle……………………………………………………………………………………………...305 Independent Events………………………………………………………………………………………………………...…308 Dependent Events……………………………………………………………………………………………………………..312 Theoretical vs. Experimental………………………………………………………………………………………………….315 Compound Events…………………………………………………………………………………………………………….321 Odds and Probability………………………………………………………………………………………………………….326 Geometric Probability………………………………………………………………………………………………………...328

Thank you for your donation to Algebra-class.com. I hope the examples included in this e-book allow you to excel in Algebra! If you find that you are still having difficulty, you may consider purchasing a unit from the “Algebra Class E-course” series. The Algebra E-courses are packed full of practice problems with a detailed, step-by-step answer key! You will also find a Chapter Test that mimics high school assessments and standardized tests! You can purchase the Algebra E-courses on Algebra-class.com If you have any other questions or concerns, please feel free to contact me! I wish you the best of luck in your studies,

Karin Hutchinson Karin Hutchinson Karin Hutchinson Karin Hutchinson


Algebra Readiness Test

Simplify: 1. 3(5+2) -8 +2(3) = 2. 10 + 9/3 • 4 +2 = 3. 33 + 52 = 4. (72 -5) • 2 = 5. 32 + 7(2+3) -1 = Simplify (If your answer is an improper fraction, rewrite it as a mixed number):

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Simplify: 11. 12(-5) = 12. -90 ÷ (-9) = 13. -15 – (-6) = 14. 22 + (-8) = 15. (-3)2 +8 -10(-2) =


Simplify: 16. 2x +3 – x = 17. 2y +3y -2 +5 = 18. 3x +2y -5 +3y -2x = 19. 3xy +2x +5y – x = 20. 3(x+1) -2x = Evaluate the expression for the given values: 21. x +y –z for x = -3 y = 2 z = -9 22. 2x +2y = for x = -2 y = 5 23. xy – 3x = for x =4 y = -8 24. Find the area of the triangle. 3.5 in 5.6 in 25. Identify the diameter of the circle. Then find the circumference. 11 cm 26. Find the perimeter and area of the square. 16 in.


27. John is going to fertilize his lawn. His lawn is a rectangle that measures 240 feet by 82 feet. The amount of fertilizer required is 0.03 ounces per square foot. •Find the area of the lawn. •How much fertilizer does John need to buy? 28. Plot the following points on the grid below. (5, 2) (-4, 1) (0, 0) (-3, -5) (1, -7)

The circle graph below shows how an average family spends money on vacation. Use this information to answer the questions below. Entertainment 24% Food 17% 55% Accommodations Gas 4%

29. If a family spends an average of $3000 on a local vacation, how much money was spent on gas? 30. What fraction of the cost is spent on food and entertainment?


Algebra Readiness Test Answer Key

Basic Math Operations

1. 19

2. 24

3. 52

4. 88

5. 43

Click here for a review on order of operations.

Fractions

6. 11/15

7. 4/15

8. 5/9

9. 1 - 1/2

10. 19/24

Click here for a review of fractions.


Integer Operations

11. -60

12. 10

13. -9

14. 14

15. 37

For examples and practice using integers, visit Basic Algebra on Algebra-class.com.

Variables - Combining Like Terms

16. x+3

17. 5y + 3

18. x +5y -5

19. 3xy +x +5y

20. x+3

For examples and practice on Combining Like Terms , visit Simplifying Algebraic Expressions on Algebra-class.com.


Evaluating Expressions

21. 8

22. 6

23. -44

Formulas

24. 9.8 in2

25. diameter = 22 cm Circumference = 69.08cm

26. Perimeter = 64 in. Area = 256 in2

27. The area of the lawn is 19680 ft2 John needs to buy 590.4 ounces of fertilizer.

Click here to revisit basic math formulas and to see examples.


The Coordinate Grid

For examples and practice using the coordinate grid visit Graphing Equations on Algebra-class.com.

Analyzing Graphs

29. A family spends $120 on gas.

30. 41/100 of the cost was spent on food and entertainment.<.p>

So, how did you do on the Algebra Readiness Test? If you did well, Congrats! If you didn't do so well, make sure you review the skills. Use the links I've provided you to find review material!


Integers and Absolute Value

Definition: The term integer represents all natural numbers and their opposites. Fractions and decimals are not integers. Integers can be shown as a set of numbers, as in this example:

Or integers can be represented on a number line as in this example:

If a number is greater than 0, it is called a positive integer. No sign is needed to indicate a positive integer. Look at the number line above. All numbers to the right of 0 are positive. Examples: 4, 8, and 15 are positive numbers. The arrows on the end of the number line indicate that the number line goes on until infinity (forever). Positive integers are pretty easy at this point, because you've been working with them since you started school! So... let's move on.

For every positive integer there is a negative integer. They are called opposites. Examples:

5 and -5 are opposites

20 and -20 are opposites


Negative Integers

Negative Integers are less than 0 (or to the left of 0 on the number line.) If a number is negative, there will be a negative (-) symbol in front of the number. Example: -2, -6, and -20 are negative integers The trickiest thing to understand about negative numbers is that the farther away from 0, the smaller the number. Example: -5 is less than -2. (Think about this - if you OWE someone $5, then you will have less money after paying, than if you only OWED $2. Other Examples: -10 is less than -8. Or... you could say that -3 is greater than -19. TIP: Think of it this way- The closer the negative number is to being positive, the larger it is!

Absolute Value

Definition: Absolute Value is the value of the number without regards to its' sign. The value is ALWAYS a positive number! The absolute value symbol is shown with the following symbol: |n| where n is any number.

Examples: |7| = 7 |-5| = 5 |-10| = 10 TIP: Your answer is just the number within the absolute value sign. Do not bring the sign with it!


Adding Integers

There are two rules that you must follow when adding integers. You must look at the signs of each number that you are adding to determine which rule to use!

Rule #1: When adding signs that are the same, ADD and

keep the sign.

Examples

Negative + Negative = Negative -3+(-2) = -5 I added the numbers 3+2 to get 5, and then

kept the negative sign, so my answer is -5.

-8+(-2)=-10 I added the numbers 8+2 to get 10, and then

kept the negative sign, so my answer is -10.

Ok... pretty easy stuff, so far! Let's see what happens when the signs are different!

Rule #2: When adding signs that are DIFFERENT, SUBTRACT the

numbers and keep the sign of the number with the LARGEST

ABSOLUTE VALUE


Examples

4+(-2) = 2 I SUBTRACTED 4-2 to get 2 and then kept the

sign of the 4 (positive) because it has the

larger absolute value.

-7+2 = -5 I SUBTRACTED 7-2 to get 5 and then kept the

sign of the -7 (negative) because it has the

largest absolute value.

8+(-9) = -1 I SUBTRACTED 9-8 to get 1 and then kept the sign of the -9 (negative) because it has the


-5+11 = 6 I subtracted 11-5 to get 6 and then kept the

sign of the 11(positive) because it has the



Subtracting Integers

There is only one rule that you have to remember when subtracting integers! Basically, you are going to change the subtraction problem to an addition problem.

RULE: When SUBTRACTING integers remember to ADD the

OPPOSITE.

TIP: For subtracting integers only, remember the phrase "Keep - change - change"

What does that mean? Keep - Change - Change is a phrase that will help you "add the opposite" by changing the subtraction problem to an addition problem. Keep the first number exactly the same. Change the subtraction sign to an addition sign. Change the sign of the last number to the opposite sign. If the number was positive change it to negative OR if it was negative, change it to positive.

Example 1

Here's an example for the problem: 12 - (-6) = ?

Keep 12 exactly the same. Change the subtraction sign to an

addition sign. Change the -6 to a positive 6. Then add and

you have your answer!

-23 - (-11) = __ + ___ = ___

= -23 + 11 = -12

14 - (-8) = __ + ___ = ___

=14 + 8 = 22


One Easy Rule for Dividing and Multiplying Integers!

This last rule for dividing and multiplying integers is the easiest of all! One simple rule to remember regarding the sign and everything else is just as you learned it in 4th grade!

If the signs are the SAME, then the answer is POSITIVE.

If the signs are DIFFERENT, then the answer is NEGATIVE.

That's it - multiply and divide as you normally would and then apply these rules to determine the sign! Ready for a few examples?

Examples

1. 7(-6) = -42 The signs are different (positive 7 and

negative 6), so the answer is negative.

2. -4(-5) = 20 The signs are the same (negative 4 and

negative 5), so the answer is positive.

3. -4/2= -2 The signs are different (negative 4 and positive 2), so the answer is negative.


Using the Distributive Property

The distributive property is used A LOT in Algebra! It is essential that you MASTER this skill in order to solve algebraic equations. First we are going to make sure that you have the proper background knowledge, so let's look at a simple math problem:

2( 3+4) = ? 2( 7) = 14

Since this problem contains all numerals, we can use the order of operations and solve inside the parenthesis first. We know that 3 + 4 = 7 and then multiply 7 by 2 and we get our answer of 14! That's basic math. But, in Algebra we often don't know one of the numeric values and a variable (letter) is used in its place. Let's look at another example:

2(x+4) = ? In this problem, we don't know the value of x, so we can't add x + 4. But... we do want to simplify it a little further by removing the parenthesis. So, this is when the distributive property comes in! To prove that the distributive property works, look at the model below.

The problem 2(x+4) means that you multiply the quantity (x +4) by 2. You could also say that you add x+4, 2 times which is the way it is shown in the model. You can see we got an answer of 2x +8 using the models. We would get the same answer using the distributive property!


Let's take a look.

Here's a couple of other examples for you to study. Copy the examples onto your paper and do them with me.

More Examples


TIP: When you simplify algebraic expressions, you DO NOT want two math symbols following each other! For example: -3x + (-4) is incorrect. See how the plus sign is followed by the negative sign? This is better read as -3x - 4. -3x - (-4) is incorrect. See how the minus sign is followed by the negative sign? This is better read as -3x + 4. If you have two math symbols following each other, change the math sign to its opposite (plus to minus or minus to plus) and change the negative sign following to a positive! Subtracting a negative is the same as adding a positive. Adding a negative is the same as subtracting a positive.


Simplifying Algebraic Expressions

There are two things that you must be able to do when simplifying algebraic expressions. The first is to be able to use the distributive property. The second math concept that you must understand is how to combine like terms. So, what are like terms? Let's start first with just the word, term and a few other vocabulary words!

2x, 5y, 7, 3xy, 4, 8x, 6y, 4x2 Take a look at the 8 terms above. (2x is a term, 5y is a term, 7 is a term...) The 4 and the 7 do not have a variable. They are called constants. Since they don't have a variable, their values will always remain the same, 4 and 7. That's why they are called constants. The 2 in the term 2x is called a coefficient. A coefficient is a number by which a variable is multiplied. The 5 in the term 5y is a coefficient; The 3 in the term 3xy is a coefficient. Can you guess which other terms have coefficients? Yes, you are right, the 8 in 8x, 6 in 6y, and the 4 in 4x2. Ok.. now moving onto like terms. Like terms are two or more terms that have EXACTLY the same variables. (The coefficients do not have to be the same, just the variables!) For example, for the 8 terms above, 2x and 8x are like terms because they both just contain an x. Can you find the other two sets of like terms? Yes, 5y and 6y are like terms. 7 and 4 are like terms because they are both constants. 3xy does not have a like term because no other term has the variable x and y. 4x2 does not have a like term because no other term has x2. Ok, enough vocabulary... let's look at a few examples. We are going to simplify each expression by combining like terms. TIP: When you combine like terms, you MUST take the sign in front of the term with it or your answer may be incorrect!


Ok, are you confident enough to use the distributive property when simplifying algebraic expressions? Sure you are, let's go! If you see an addition or subtraction problem inside a set of parenthesis, you must use the distributive property BEFORE simplifying the expression. As you review the next 2 examples, notice how the distributive property was used first, then the algebraic expression was simplified.


Ok... with four great examples, you should be ready to try some on your own!


Balancing Equations

Before we get into the meat of our Solving Equations unit, we are going to quickly take a look at balancing equations. As you begin solving equations in Algebra, the very first thing that you will learn is that whatever you do, you MUST keep your equation balanced! It truly is the number one factor in solving equations! You will see step-by-step how equations are kept balanced for every example in this unit! In the explanations, you will see the words:

"Whatever you do to one side, you must do to the other side!" Those are the key words for balancing equations! Let's take a look at a graphic to give you a better understanding.

Think of a scale that you would use to weigh things! If you put a 5 lb weight on the left side, what happens? Is the scale balanced? NO... of course not! What would you have to do to balance it perfectly? You'd have to put another five pound weight (or something that weighs five pounds) on the other side. So... if I put 5 lbs one side, then I MUST also put 5 lbs on the OTHER side! That's the only way it will stay balanced! So... does this phrase make sense...?

"Whatever you do to one side, you must do to the other side!"


Think of the equation sign as the "center" of the scale. Take a look at the picture above, and imagine two 5 lb weights on the left side, and two 5 lb weights on the right side. How much weight is on each side? 10 lbs, right? So, right now the scale is balanced! Now, what happens if I take 5 lbs off of the right side? Is the scale balanced now? No, there's 10 lbs still on the left, but only 5 lbs on the right. So, what must I do to keep it balanced? Yes.... I must ALSO take 5 lbs off of the left! Now, it is balanced again! . So, in Algebra if you subtract 5 from one side, you must also subtract 5 from the other side. Again....

"Whatever you do to one side, you must do to the other side!"

Why have I continued to repeat those same words? Because... those are the words that you MUST remember with EVERY step of EVERY equation that you solve! You will see those words in all of my explanations, so look for them! Keep those equations balanced and you will do well!


Solving One Step Equations Addition Equations

There are four things that you must remember when solving any equation AND everything must be mathematically correct!

• Your goal is to get the variable that you are solving for by itself on

one side of the equation.

Example: x = _____

• To eliminate a term on one side of the equation, you use the opposite

mathematical operation.

• Whatever you do to one side of the equation, you MUST do to the

other side of the equation. This keeps the equation balanced!

• Always check your answer by substituting the value back into the

original equation.


Here we have two more examples of how to solve an addition equation! I've even thrown in a few negatives, because that can sometimes confuse you, especially if you don't remember your addition integer rules or your subtraction integer rules.


Solving One-Step Equations with Subtraction

Now we are going to be solving one-step equations that involve subtraction! We are going to use the same methods to solve as we did with equations that were addition problems. Except this time, can you guess what the opposite mathematical operation is going to be? You got it - Addition! We are going to add in order to get the variable by itself! Here are a few examples!

Example 1


Example 2


Example 3


Solving One-Step Equations Multiplication Equations

The next set of one-step equations do not contain a constant that you must add or subtract to remove. These equations contain a coefficient. A coefficient is a number that is multiplied by the variable. Therefore, we must remove the coefficient! Take a look at this equation: 3x = 9. Since there is no mathematical symbol between the 3 and the x we know that means multiplication. So, what number times 3 will give us an answer of 9? You know the answer, right? Yes, 3! 3*3 = 9! Another question to ponder- What is the opposite of multiplication? Yes... division! We are going to divide in order to get x by itself! Why divide? What is 3/3? Yes... 1! What is 1*x? You got it.... x! That's how we get x by itself! We want the coefficient to be 1. Anytime you divide a number by itself, you will get an answer of 1! Let's look at a few examples:

Example 1


Example 2

Not too hard if the coefficient is an integer! What happens if the coefficient is a fraction?

Think: If I have 2/3x, how can I make 2/3 a coefficient of 1? Yes... you will divide by 2/3, but...Do you remember what a reciprocal is in math? When you divide a fraction, you actually multiply by the reciprocal. If you take 2/3 and you flip it to 3/2, that is the reciprocal! If you multiply by the reciprocal, you will have a coefficient of 1. Watch this:


Example 3

You may also have problems where your answer results in a fraction. Here's one more example:


Example 4


Solving Basic Equations

Equations with Division Ok, last set of equations before we pull it all together to solve two-step equations. You can probably guess now, what mathematical operation we will use to "undo" a division equation. Yes... Multiplication! We are actually going to multiply by the value of the denominator!


Solving Two-Step Algebra Equations

Now that you know all the rules for solving one-step equations, solving two-step algebra equations will be a piece of cake! You will need prior knowledge of solving one-step equations in order to understand this lesson. Click here to complete the solving one-step equations lesson. Each equation is going to be solved in two separate steps. Let's take a look:

2x + 3 = 43 Notice that in order to get x on the left hand side by itself, (x = ) we need to remove the 3 and the 2. Therefore, this equation will involve two separate steps. Remember that you must use the opposite mathematical operation in order to remove a number from one side of an equation. There's one rule to remember when solving two-step equations:

Always remove the constant first using addition or

subtraction.

Your second step will be to remove any coefficients or

divisors using multiplication or division.

Let's solve this example together. Notice how two separate steps are involved in solving this equation!


Example 1

We'll look at another example similar to this one.


Example 2

The next two examples will demonstrate how to solve two-step equations when the coefficient is a fraction or when you have a divisor. Let's take a look!


Example 3


Example 4

Ok... one more example! This example involves an additional step before starting the two-step process. .If you have any terms in the equation that are like terms, then you will want to combine those like terms first before


solving. Let's take a look....

Example 5


Using the Distributive Property

The distributive property is used A LOT in Algebra! It is essential that you MASTER this skill in order to solve algebraic equations. First we are going to make sure that you have the proper background knowledge, so let's look at a simple math problem:

2( 3+4) = ? 2( 7) = 14

Since this problem contains all numerals, we can use the order of operations and solve inside the parenthesis first. We know that 3 + 4 = 7 and then multiply 7 by 2 and we get our answer of 14! That's basic math. But, in Algebra we often don't know one of the numeric values and a variable (letter) is used in its place. Let's look at another example:

2(x+4) = ? In this problem, we don't know the value of x, so we can't add x + 4. But... we do want to simplify it a little further by removing the parenthesis. So, this is when the distributive property comes in! To prove that the distributive property works, look at the model below.

The problem 2(x+4) means that you multiply the quantity (x +4) by 2. You could also

say that you add x+4, 2 times which is the way it is shown in the model. You can see we got an answer of 2x +8 using the models.

We would get the same answer using the distributive property! Let's take a look.


Here's a couple of other examples for you to study. Copy the examples onto your paper and do them with me.

More Examples


TIP: When you simplify algebraic expressions, you DO NOT want two math symbols following each other!

For example: -3x + (-4) is incorrect. See how the plus sign is followed by the negative sign? This is better read as -3x - 4. -3x - (-4) is incorrect. See how the minus sign is followed by the negative sign? This is better read as -3x + 4. If you have two math symbols following each other, change the math sign to its opposite (plus to minus or minus to plus)and change the negative sign following to a positive! Subtracting a negative is the same as adding a positive. Adding a negative is the same as subtracting a positive.


Solving Equations with Fractions

Do you start to get nervous when you see fractions? Do you have to stop and review all the rules for adding, subtracting, multiplying and dividing fractions? If so, you are just like almost every other math student out there! But... I am going to make your life so much easier when it comes to solving equations with fractions! Our first step when solving these equations is to get rid of the fractions because they are not easy to work with! Let's look at an example:

That makes sense, right? We don't want to solve from here and end up having to subtract 9/4 from 9. We also don't want to multiply by the reciprocal yet, because we have so many terms, that again it will create more fractions within the problem. So... what do we do? We are going to get rid of just the denominator in the fraction, so we will be left with the numerator, or just an integer! I know, easier said than done! It's really not hard, but before I get into it, I want to go over one algebra definition. We need to discuss the word term. In Algebra, each term within an equation is separated by a plus (+) sign, minus (-) sign or an equals sign (=). Variable or quantities that are multiplied or divided are considered the same term.


For example:

That last example is the most important to remember. If a quantity is in parenthesis, it is considered one term! Let's look at a few examples of how to solve these crazy looking problems!


Example 2


Solving Literal Equations

I know that in your Math studies, you have come across numerous formulas. Most of these formulas have probably involved geometry! Have you ever been given a formula, and needed to solve for a variable within the formula? For example, let's take the distance formula. D = rt. Let's say that you know the distance is 50 ft and the time to travel was 5 minutes. You need to find the rate travelled. In this case, you need to solve for a variable within the formula (rate), and not the standard, (distance). Yes, I know these problems are always a little more difficult. I'm going to show you a step that will make this problem easier to solve. We are going to be solving literal equations and this means that we will be solving a formula for a given variable. We are going to use all of the rules that we've learned for solving equations to solve literal equations. You will need to perform "opposite operations" and whatever you do to one side of the equation you must do to the other side of the equation! Let's look at a couple of examples!


Example 1

In this example, we'll use the distance formula that we talked about earlier!

Now let's look at an equation that involves fractions.


Example 2


Solving Algebra Equations with Variables on Both Sides

This is it! Our last type of equation to solve before applying our knowledge to solve real world problems! Yes... the whole reason we are learning to solve these equations! So far, you've learned how to solve one-step equations, two-step equations, and equations with fractions. If you haven't mastered these skills or need a refresher, please go back and review these lessons before attempting this one! It will be so much easier if you have the background! So, if you are ready, let's move on! Equations with variables on both sides look intimidating, but they really aren't that hard if you have the background skills. In fact, I think these types of equations are fun because it's like moving pieces of a puzzle around. This is what you need to remember:

• If you have any fractions, get rid of those first by

multiplying ALL terms by the denominator.

• Use the distributive property if needed.

• Your ultimate goal is to get all of the constants on one side

of the equation and all of the variables on the other side of

the equation. You can accomplish this by adding or

subtracting terms on BOTH sides of the equation.

Let's look at a few examples.


Example 1

This next example shows how to solve if the distributive property is involved.


Example 2

This last example shows how to solve if fractions are involved.


Example 3

It doesn't look too hard, does it? Remember: it's like moving puzzle pieces around - all the variable terms on one side and all the constant terms on the other side!


REAL WORLD PROBLEMS: How to Write Equations Based on Algebra Word Problems

I know that you often sit in class and wonder, "Why am I forced to learn about equations, Algebra and variables?" But... trust me, there are real situations where you will use your knowledge of Algebra and solving equations to solve a problem that is not school related. And... if you can't, you're going to wish that you remembered how. It might be a time when you are trying to figure out how much you should get paid for a job, or even more important, if you were paid enough for a job that you've done. It could also be a time when you are trying to figure out if you were over charged for a bill! This is important stuff - when it comes time to spend YOUR money - you are going to want to make sure that you are getting paid enough and not spending more than you have to! Ok... let's put all this newly learned knowledge to work. Click here if you need to review how to solve equations. There are a few rules to remember when writing Algebra equations:

• First, you want to identify the unknown, which is your variable. What

are you trying to solve for? Identify the variable: Use the statement,

Let x = _____. You can replace the x with whatever variable you are

using. • Look for key words that will help you write the equation. Highlight

the key words and write an equation to match the problem. • The following key words will help you write equations for Algebra

word problems: 1. more than - means add

2. less than - means subtract

3. times as many - means multiply

4. "per" - means multiply


The next example shows how to identify a constant within a word problem.


The last example is a word problem that requires an equation with variables on both sides.


Using a Table of Values to Graph Linear Equations

You can graph any equation using a table of values. A table of values is a graphic organizer or chart that helps you determine two or more points that can be used to create your graph.

Why Use a Table of Values?

In order to graph a line, you must have two points. For any given linear equation, there are an infinite number of solutions or points on that line. If you just find two of the solutions, then you can plot your two points and draw a line through. This will be the line that represents the equation. Every point on that line is a solution to the equation. In my table, I have 4 columns as described below:

• The first column is for the x coordinate. For this column, I can choose any number I wish. Try to choose numbers that can be graphed on your graph. For example, if your x axis only extends to 10, don't choose 12 as an x coordinate.

• The second column is for substituting x into the equation in order to solve for y. So, whatever value I chose for x, I will substitute back into the equation and solve to find the y value.

• The third column is for the y value. After substituting your x value into the equation, your answer is the y coordinate.

• The last column is for your ordered pair. Your ordered pair is the x value and the y value. This is the point on your graph. Let's look at a few examples, and it will all make more sense!


• In this first example, I chose -2, 0, and 2 as my x coordinates.

• After substituting those values into the equation: y = 2x +1, I found my y values to be: -3,

1, and 5.

• Therefore, the ordered pairs that I found on my graph were: (-2,-3), (0,1), and (2,5).

• I plotted those points on my graph.

• I then used my ruler and drew a straight line through those points. This is the line for the

equation, y = 2x +1.

• If you had done this problem on your own, you may have found three different points

using the table of values. That's ok, because even if your three points are different, your

line will still look exactly the same!


• We can also find other solutions for the equation just by reading the graph. I see that

(3,7) is a point on the graph. If I substitute 3 for x into the equation, I will get 7 as my y

coordinate.

• This line goes on forever, so there are infinite solutions to the equation.

Let's look at another example.


• This equation, y = -1/2 x - 1 has a fraction as the coefficient of x. This becomes a little

trickier in choosing x coordinates because we could end up with a fraction for the y

coordinate. This then becomes a little more difficult to graph.

• So, the trick is to look at the denominator of the coefficient. You want to choose x

coordinates that are either multiples of the denominator or 0. 0 is the easiest choice.

Then choose either the same value as the denominator or a multiple of the

denominator.

• Remember, you also have a choice of positive or negative numbers! This will ensure

that your y coordinate is an integer which is much easier to graph.

Tip

If the coefficient is a fraction, choose 0 or a multiple of the denominator as your x coordinates!

Ok, one more example:


• This example has a slightly different direction, but involves the same process.

• The problem asks for 3 solutions.

• Remember, that when you find ordered pairs in your table of values, these are actually

solutions to the equation.

• There are other solutions, which are all of the other points on the line.

• Any point on the line would be a correct answer to this problem.


Calculating Slope The Key to Graphing Equations

What do you think of when you hear the term, slope? Do you think of a skier, skiing down a large mountain? Or, maybe you think of the sliding board at the playground. Whatever you are thinking, it's probably something that's on an incline! When we study slope in Algebra we are going to study the incline and other characteristics of a line on a graph. Slope is a very important concept to understand in Algebra. Therefore, I've created three different lessons to help you gain a full understanding. We will start here with defining and calculating slope by analyzing a graph. Then we will move on to graphing slope and finally to using slope intercept form to create your graph. Start here from the beginning, or move onto the concept of slope that you need help with!

What is Slope in Algebra?

Slope is used very often in Mathematics. It can be used to actually find how steep a particular line is, or it can be used to show how much something has changed over time. We calculate slope by using the following definition. In Algebra, slope is defined as the rise over the run. This is written as a fraction like this:


Rules for Calculating the Slope of a Line

• Find two points on the line.

• Count the rise (How many units do you count up or down to get from one point to the next?) Record this number as your numerator.

• Count the run (How many units do you count left or right to get to the point?) Record this number as your denominator.

• Simplify your fraction if possible.

IMPORTANT NOTE:

• If you count up or right your number is positive.

• If you count down or left your number is negative.

Let's look at an example:


Graphing a Linear Equation Using Slope Intercept Form

Now that you've completed a lesson on graphing slope you are finally ready to graph linear equations.

There are several different ways to graph linear equations. You've already learned how to graph using a table of values. That's okay for the beginner, but it can be a little time consuming.

Using slope intercept form is one of the quickest and easiest ways to graph a linear equation.

Before we begin, I need to introduce a little vocabulary. We are going to talk about x and y intercepts.

An x intercept is the point where your line crosses the x-axis. The y intercept is the point where your line crosses the y-axis.

We are only going to focus on the y intercept in this lesson, but you'll need to know x intercept for later.

Let's look at an example:


Slope intercept form is used when your linear equation is written in the form:

y = mx+b

x and y are your variables. m will be a numeral, which is your slope. b will also be a numeral and this is the y-intercept.

In this form only (when your equation is written as y = ....) the coefficient of x is the slope and the constant is the y intercept.

Let's look at a few examples and I promise that you'll LOVE this new way of graphing!


Example 1


Example 2


• Your y intercept is always the first point that you plot on the line. Your point will always be (0, b).

• Then use your slope to plot your next point.

• If you have two points, you can draw a straight line and this is the line that represents your equation. Any point on that line is a solution to the equation.

Tip: You have to be very accurate in plotting your points and drawing your lines in order to be able to read your graph to find other solutions!

Here's a quick summary to help remind you of the steps for graphing in slope intercept form.

Rules for Graphing Using Slope Intercept Form


Rate of Change

Connecting Slope to Real Life Why do we need to find the slope of a line? The slope of a line tells us how something changes over time. If we find the slope we can find the rate of change over that period. Take a look at the following graph.

This graph shows how John's savings account balance has changed over the course of a year. We can see that he opened his account with $300 and by the end of the first month he had saved $100. By the end of the 12 month time span, John had $1500 in his savings account. John may want to analyze his finances a little more and figure out about how much he was saving per month. This is called the rate of change per month. By finding the slope of the line, we would be calculating the rate of change. We can't count the rise over the run like we did in the calculating slope lesson because our units on the x and y axis are not the same. In most real life problems, your units will not be the same on the x and y axis. So, we need another method! We will need to use a formula for finding slope given two points.


Finding Slope Given Two Points


Rate of Change and Real Life Problems

Let's go back and look at John's Savings Account graph again.


And... one last example.


The three examples above demonstrated three different ways that a rate of change problem may be presented. Just remember, that rate of change is a way of asking for the slope in a real world problem. Real life problems are a little more challenging, but hopefully you now have a better understanding.


Graphing Linear Equations in Standard Form You have learned many techniques for graphing linear equations that are written in slope intercept form! Yes... if the equation is already written in slope intercept form, graphing is pretty easy! However, sometimes you will see equations that are written in standard form.

What is Standard Form?

If the equation is presented in standard form, then you are not able to identify the slope and y-intercept that are needed for graphing! So... what should you do? There are actually two different techniques that you can use for graphing linear equations that are written in standard form! You can use either method, so I'm going to demonstrate both methods and you may come to favor one over the other! In this particular lesson, we are going to study how to convert the standard form equation into an equation written in slope intercept form! In the next lesson, we will study a method using x and y intercepts! Let's get started! If you have a standard form equation, you can rewrite it in slope intercept form! Let's look at a couple of examples!


Example 1

Let's look at one more example!


Example 2

This shouldn't be too hard, since you've already mastered the skills for solving equations and the skills for graphing in slope intercept form!


Using the X and Y Intercept to Graph Linear Equations

You've learned one way to graph a standard form equation - by converting it to slope intercept form! Click here to review this lesson! There is another way to graph standard form equations, and that is to find the x and y intercepts! Before we begin, let's quickly review what standard form looks like!

What is Standard Form?

Now let's review what the term intercepts means! An intercept is where your line crosses an axis. We have an x intercept and a y intercept. The point where the line touches the x axis is called the x intercept. The point where the line touches the y axis is called the y intercept. Take a look at the graph below.


If we can find the points where the line crosses the x and y axis, then we would have two points and we'd be able to draw a line. When equations are written in standard form, it is pretty easy to find the intercepts. Take a look at this diagram, as it will help you to understand the process.


To find the X Intercept: Let y = 0 To find the Y Intercept: Let x = 0

Example 1

Ok.. now let's look at a real world problem that we can solve using intercepts.


Example 2


Writing Equations in Slope Intercept Form

Let's first quickly review slope intercept form.

Equations that are written in slope intercept form are the easiest to graph! If you need to create a graph, this is a skill that is essential! Continue reading for a couple of examples!

Example 1

Write the equation for a line that has a slope of -2 and y-intercept of 5.

NOTES: I substituted the value for the slope (-2) for m and the value for the y-intercept (5) for b. The variables x and y should always remain variables when writing a linear equation.

That's pretty easy, now let's look at a graph.


Example 2

Write an equation for the following line:

m = 3 b = -2

y = mx+b

y = 3x -2

NOTES: As I analyze the graph, I notice that the line crosses the y axis at the point, (0,-2). Therefore, my y-intercept

is -2.

I then count the slope from the y-intercept to another point on the line. I count up 3 and right 1. 3/1, which means

that my slope is 3. This is the value for m.

Just to double check that my slope is positive, I notice that as I follow the line from left to right, the line is rising. Therefore, my slope is positive!

The equation for this line is: y = 3x -2


Let's look at another graph.

Example 3

Write an equation for the following line:

m = -3/4 b = 3

y = mx+b

y = -3/4x +3

NOTES: As I analyze the graph, I notice that the y-intercept (y value of the point where the line crosses

the y axis) is 3. So 3 is my value for b.

I then count the slope from the y-intercept to another point on the line. I count down 3 and right 4

(or I could count up 3 and left 4). This means that my slope is -3/4. This is the value for m.

Just to double check that my slope is negative, I notice that as I follow the line from left to right, the line is falling. Therefore, my slope is negative!

The equation for this line is: y = -3/4x + 3


When you have a real world (word problem) that requires you to write an equation in slope intercept form, there are two things that you want to look for:

Use the chart below to help you organize your information as you analyze each word problem. This will help you to write your equation!

Take a look at the examples below to better clarify how this chart can help you!


Example 4


Example 5


Writing Equations in Standard Form

We know that equations can be written in slope intercept form or standard form.

Let's quickly revisit standard form. Remember standard form is written:

Ax +By= C

We can pretty easily translate an equation from slope intercept form into standard form. Let's look at an example.

Example 1

That was a pretty easy example. We just need to remember that our lead coefficient must be POSITIVE!


Let's take a look at another example that involves fractions! There is one other rule that we must abide by when writing equations in standard form.

Equations that are written in standard form: Ax +By = C

CANNOT contain fractions or decimals! A, B, and C MUST be integers!

Let's take a look at an example!

Example 2

Now, let's look at an example that contains more than one fraction with different denominators!


Example 3

Slope intercept form is the more popular of the two forms for writing equations. However, you must be able to rewrite equations in both forms.

For standard form equations, just remember that the A, B, and C must be integers and A cannot be negative!


Solving Word Problems in Algebra Writing Standard Form Equations

We've studied word problems that allow for you to write an equation in slope intercept form. How do we know when a problem should be solved using an equation written in standard form?

As you are reading and analyzing the word problem, if you find that you can set up an addition problem, and you have a set total (constant), then you will be able to write an equation in standard form.

Let's look at a couple of examples!

Example 1


Let's look at one more example.

Example 2

I know it's hard to imagine when you will need to use this skill, or even how to solve problems written in this form! You will be writing these equations and solving them when you get to the Systems of Equations Unit. So, stick with me - it will all come together and make sense!


Writing Linear Equations Given Slope and a Point

When you are given a real world problem that must be solved, you could be given numerous aspects of the equation. If you are given slope and the y-intercept, then you have it made!

You have all the information you need, and you can create your graph or write an equation easily.

You may be given slope and a point (not necessarily the y-intercept). This then becomes a little trickier to write an equation, because although you have the slope, you need the y-intercept.

You have enough information to find the y-intercept, but it requires a few more steps! Let's look at an example.


Example 1


Great! Now let's look at real world applications of this skill!


Ok, so if you are given slope and a point, then you need to substitute for m (slope), x, and y and then solve for b! Once you have m (slope) and b (y-intercept), you can write an equation in slope intercept form!


Writing Algebra Equations Finding the Equation of a Line Given Two Points

We have written the equation of a line in slope intercept form and standard form. We have also written the equation of a line when given slope and a point. Now we are going to take it one step further and write the equation of a line when we are only given two points that are on that line.

This type of problem requires three steps. We must first use our information to find the slope and then use the slope and a point to find the y-intercept.

REMEMBER: If I know the slope and y-intercept, then I can write an equation in slope intercept form. Once the equation is in slope intercept form, I can rewrite it in standard form if I need to!

Let's look at an example.


Ok, now let's apply this skill to solve real world problems!

Writing Algebra Equations Real World Problems


Graphing Systems of Equations

This is the first of four lessons in the System of Equations unit. We are going to graph a system of equations in order to find the solution. REMEMBER: A solution to a system of equations is the point where the lines intersect! Prerequisites for completing this unit: Graphing using slope intercept form .

We will begin graphing systems of equations by looking at an example with both equations written in slope intercept form. This is the easiest type of problem!

Example 1


Now we are going to look at a system of equations where only one of the equations is written in slope intercept form. The other equation is written in standard form. So... what do you think we need to do first?

Example 2


Example three has no solution.


Example 3

Tip:

Whenever two equations have the same slope they will be parallel lines. Parallel lines NEVER intersect. Therefore, the system of equations will NOT have a solution!

Our last example demonstrates two different things. The first is that there is more than one way to graph a system of equations that is written in standard form. The second is that sometimes a system of equations is actually the same line, graphed on top of each other. In this case, you will see an infinite number of solutions! It may be helpful for you to review the lesson on using x and y intercepts for this example. Check it out!


Example 4


Did you notice that both equations had the same x and y intercept? This is because these two equations represent the same line. Therefore, one is graphed on top of the other! In this case, the system of equations has an infinite number of solutions! Every point on the line is a solution to both equations. Now let's look at the other method for solving this system of equations. I am showing you both methods to remind you that you do have choices when solving a system of equations. There is more than one way to reach the solution.


Now you've seen two different ways to graph a system in standard form AND you've seen what happens when the equations are actually the same line!


Using the Substitution Method to Solve Systems of Equations

The substitution method is one of two ways to solve systems of equations without graphing. The following steps can be used as a guide as you read through the examples for using the substitution method.

Steps for Using the Substitution Method in order to Solve Systems of Equations

• Solve 1 equation for 1 variable. (Put in y = or x =

form)

• Substitute this expression into the other equation and

solve for the missing variable.

• Substitute your answer into the first equation and solve.

• Check the solution.

These directions will make a lot more sense when you study the examples below!


Let's take a look at another example. You'll find this very interesting!

Ok... one more unique example!


Solving Systems of Equations Using Linear Combinations

There are two ways to solve systems of equations without graphing. You can use the substitution method or linear combinations. This lesson is going to focus on using linear combinations. The following steps are a guide for using Linear Combinations. Don't worry, it will make a lot more sense as we look at a few examples.

Steps for Using Linear Combinations

• Arrange the equations with like terms in columns.

• Analyze the coefficients of x or y. Multiply one or both equations by

an appropriate number to obtain new coefficients that are opposites.

• Add the equations and solve for the remaining variable.

• Substitute the value into either equation and solve.

• Check the solution.

Ok... let's make these examples make sense by looking at some examples.


The next problem demonstrates the extra step that you need to take if your original problem doesn't have opposite terms! Look for that extra step!

Example 2


Let's see what happens if our system of equations happens to be the same line!

Example 4


Last example! We are going to see what happens when you try to use linear combinations to solve a system that has parallel lines!

Example 5


Solving Systems of Equations Real World Problems

Wow! You have learned many different strategies for solving systems of equations! First we started with Graphing Systems of Equations. Then we moved onto solving systems using the Substitution Method. In our last lesson we used the Linear Combinations or Addition Method to solve systems of equations. Now we are ready to apply these strategies to solve real world problems! Are you ready?

First let's look at some guidelines for solving real world problems and then we'll look at a few examples.

Steps For Solving Real World Problems

• Highlight the important information in the problem that will help

write two equations.

• Define your variables

• Write two equations

• Use one of the methods for solving systems of equations to solve.

• Check your answers by substituting your ordered pair into the

original equations.

• Answer the questions in the real world problems. Always write your

answer in complete sentences!

Ok... let's look at a few examples. Follow along with me! (Having a calculator will make it easier for you to follow along!)


Example 2


Yes, I know that word problems can be intimidating, but this is the whole reason why we are learning these skills. You must be able to apply your knowledge!


Solving Inequalities in One Variable

Are you ready to dive into our inequalities unit? Let's do a very quick review of inequality basics that you probably first started learning about in second grade! The unique thing about inequalities is that there is not just one solution, but there are multiple solutions! Think about the following inequality:

x < 5 (Read as x is less than 5)

• We could replace x with 4 because 4 is less than 5.

• We could replace x with 2 because 2 is less than 5.

• We could even replace x with -3 because -3 is less than 5. We could go on forever, so as you can see there are many solutions to this

inequality!

Let's take a look at the inequality symbols and their meanings again.


When you graph inequalities that have only one variable, we use a number line. We will use open and closed circles and arrows pointing to the left or right to graph our answers. Take a look at the model below.


Ok... enough review. Let's solve a few inequalities! You are going to solve inequalities, exactly the same as you solved equations. Click here if you need a refresher on solving equations. There is however, one small rule that you always have to remember when solving inequalities! (Yes... it's always something, isn't it?) We'll get to that in a minute. Let's look at a simple inequality first!


Example 1

The next example is similar to example 1, but I would like to show you how to reverse your answer to make it easier to read and graph! Don't be afraid to do this if your variable ends up the right hand side of the inequality!


Example 2

Ok... The first two examples should have been pretty easy since you are a superstar at solving equations!


Luckily there's only one trick that you have to remember when solving inequalities and that is:

Whenever you multiply or divide by a negative number, you must reverse the sign!

I knew you were going to ask "Why?" And.... you should be asking why! It's important to understand these rules! Sometimes, the best explanations are through examples. Let's take a look.

This works with any true statement as long as you multiply or divide by a negative number! Go ahead, try another one! Write a true statement, and then divide by -2!


Ok... let's look at a few examples! Our next example revisits how to solve equations and/or inequalities with variables on both sides!


Example 3

Our last example revisits how to solve equations and/or inequalities with fractions. I hope you remember the trick! If not, click here!


Example 4

3x > 3/5(x -2) Original Problem

5[3x > 3/5(x -2)] Multiply EVERY term by 5 to get rid of the fraction 3/5. Remember: 3/5(x-2) is all one term.

15x > 3(x-2) Simplify.

15x > 3x – 6 Distribute.

15x-3x > 3x -3x -6 Subtract 3x from BOTH sides.

12x > -6 Simplify.

12x > -6 12 12

Divide BOTH sides by 12

x > -1/2 Simplify. Don’t reverse the sign because you did not divide by a -12, you divided by a positive 12!

Check: 3x > 3/5(x -2) 3(0) > 3/5(0 -2) 0> 3/5(-2) 0≥ -6/5

I can choose -½ or any number greater than -1/2. I chose 0, and I have a true statement!

Since x is greater than or equal to -1/2, I have a closed circle on -1/2 and a line pointing to the right to indicate greater than.


Solving Word Problems in Algebra Inequality Word Problems

Are you ready to dive into the "real world" of inequalities? I know that solving word problems in Algebra is probably not your favorite, but there's no point in learning the skill if you don't apply it. I've tried to provide you with examples that could pertain to your life and come in handy one day! Think about others ways you might use inequalities in real world problems! I'd love to hear about them if you do! Before we look at the examples let's go over some of the rules and key words for solving word problems in Algebra (or any math class).

Word Problem Solving Strategies

• Read through the entire problem.

• Highlight the important information and key words that you need to

solve the problem.

• Identify your variables.

• Write the equation or inequality.

• Solve.

• Write your answer in a complete sentence.

• Check or justify your answer.

I know it always helps too, if you have key words that help you to write the equation or inequality. Here are a few key words that we associate with inequalities! Keep these handy as a reference!


Inequality Key Words

• at least - means greater than or equal to

• no more than - means less than or equal to

• more than - means greater than

• less than - means less than

Ok... let's put it into action and look at our examples!


Example 2


Graphing Inequalities

In the previous lesson, we learned how to solve and graph inequalities in one variable. We learned that an inequality has MULTIPLE solutions and we graphed those solutions on a number line. Now we are going to move onto solving and graphing inequalities in two variables. You may also know these as Linear Inequalities. Again, we are going to apply most of the same rules for graphing equations but there are going to be a few new rules that we must apply! Yes... that's what makes this challenging and exciting! You will definitely need to remember how to graph equations using slope intercept form, so if you need a refresher, please click here now! There are two things that are going to be different with inequalities. The first, is we are going to be using either a solid line or a dotted line when graphing the equation. The second is, we are going to be shading one side of the line to show all of the MULTIPLE solutions to the inequality. So... first let's take a look at our graphing symbols. You may want to keep this handy for a reference.


Don't worry, when you look at our first example, this will all make more sense!


For this second example, we'll need to rewrite the equation so that it's in slope intercept form before we graph. Also take note that the sign is greater than or equal to, so we will graph a solid line this time instead of a dotted line.


Example 2


Example 3


Graphing Systems of Inequalities

This lesson shouldn't be too hard, because we are just going to apply all of our knowledge from graphing linear inequalities to create a system of inequalities.


Not too bad, is it? It might help for you to have two different colored pencils if you are practicing along with me. If you don't have colored pencils, then you can draw horizontal lines for one inequality and vertical lines for the other. This will make it easier to see which area contains solutions for both inequalities. The next example will demonstrate how to graph a horizontal and a vertical line.


Example 2


Systems of Inequalities Word Problems

As you get further into Algebra 1, you will find that the real world problems become more complex. They have more questions to be answered and require more steps to find the solution. When you get into systems of inequalities, this is especially true because you are dealing with two inequalities. But... don't let that intimidate you! You have all the skills that you need to solve these problems. Take one step at a time and think about what the question is asking for! Read through my example very carefully, and study how I performed each step! Pay careful attention to the key words (highlighted words) and how each inequality was written based on the problem.


Introduction to Algebraic Relations and Functions

Before we jump into discussing functions, we're going to take a step back and talk about algebraic relations and a few other vocabulary words. I know that you may be anxious to get to the "algebra problems", but this page contains a lot of vocabulary that you will need to understand the remainder of the unit. So, make sure you take the time to read and understand this page before moving on! A relation is a set of ordered pairs. The first item in an ordered pair is identified as the domain. The second item in the ordered pair is identified as the range. Let's take a look at a couple of examples:

Example 1

For the first example, I'm going to show you something that you are familiar with. We all know there is a relationship between a vehicle and the number of wheels that it contains. . A relation can be written in the form of a table:

As you know, in Algebra, we will not be dealing with vehicles as a domain. I used this example just to define the vocabulary because it is something that you are familiar with. Now, I want you to think of a Vending Machine. How about the classic vending machine that you see in office buildings, stores, schools... where you can choose chips, crackers or candy. You put in 75 cents and out pops your bag of chips. Or... you put in $1.00 (Yes... aren't they getting expensive these days?) and out pops your Hershey Bar. There is a relationship between the amount of money that you put in the machine and what comes out! This is exactly what the "Math World" is like. It's a ton of little vending machines that "swallow" an input number (domain) and pops out another number (range). Let's look at a more mathematical example.


Example 2

The following is an algebraic relation that we will call b.

b:{(2,4) (3,6) (4,8) (5, 10)}

The domain is: 2, 3, 4, 5 (These are all the x values of the ordered pair) The range is: 4, 6, 8, 10 (These are all the y values of the ordered pair) The domain contains the independent variable and the range contains the dependent variable. This means that the value of the range depends on the domain. Think about the vending machine: What comes out of the machine (range) depends on what you put in (domain). You can't put in a nickel and expect a chocolate bar to pop out! Ok... enough vocabulary about Relations. Now we want to focus on a special type of relation called a Function.


Is the Relation a Function? Using the Vertical Line Test

A function is a relation (a set of ordered pairs) where the value of one variable depends on the value of the other variable. However, in a function, each input (x coordinate) may be paired with only ONE output (y coordinate). Let's look at a couple of examples to clarify this definition.

Example 1

Let's look at our relation, b that we used in our relations example in the previous lesson.. Is this relation a function? Is each input only paired with only one output? There are actually two ways to determine if a relation is a function. One way is to analyze the ordered pairs, and the other way is to use the vertical line test. Let's analyze our ordered pairs first.

Since each input has a different output, this can be classified as a function. Let's verify it with the vertical line test. The vertical line test is used when you graph the ordered pairs. You imagine a vertical line being drawn through the graph. If the vertical line only touches the graph at one point, then it is a function. If the vertical line touches in more than one point, then it is NOT a function. Let's graph our points and use the vertical line test to prove that this is a function. I drew the vertical lines (output) on the graph to demonstrate what it would look like. Unless asked, you really don't need to draw the vertical lines, you can just imagine the vertical line, or I have my students use the edge of a sheet of paper and move it across the graph.


Since each vertical line only touches the graph at one point, this relation can be classified as a function!

Let's take a look at another example.

Example 2

Is the relation s, a function?

s:{(-3, 2) (-1, 6) (1,2)

Let's ask ourselves: Is each input paired with only one output? (This one is a little tricky!) YES, each input is paired with only one output! I know that you see an output of 2 twice, but each is paired with a different input. The outputs can be the same, as long as the inputs are different! Let's look at the vertical line test.


Yes, this is a function - remember that the outputs can be duplicated as long as the inputs are different!

Ok... I hope that you aren't confused by that last example! If you are, take a look at this last example. You will see a relation that is NOT considered a function. Hopefully this will help build your understanding!

Example 3

Is the relation, c, a function? Let's ask ourselves: Is each input paired with only one output?

c:{(3,3) (-1,0) (3,-3)

NO, look at the ordered pairs - there are 2 ordered pairs with input of 3 and they have different ranges! Think about our little function machine - you cannot put in a 3 and get out a 0, and then put in another 3 and get out a -3. Each time you put in a 3, you should get the same answer! Therefore, this relation is NOT a function. Let's look at the vertical line test.


So, how do you feel about identifying whether or not a relation is a function? I know it can be a little confusing. Just remember: If you have duplicate inputs and they are paired with different outputs, then the relation is not a function. The vertical line test is always a good technique to use if you are unsure or want to justify your answer!


Function Notation

In the previous lesson, you learned how to identify a function by analyzing the domain and range and using the vertical line test. Now we are going to take a look at function notation and how it is used in Algebra. The typical notation for a function is f(x). This is read as "f of x" This does NOT mean f times x. This is a special notation used only for functions! However, f(x) is not the only variable used in function notation! You may see g(x), or h(x), or even b(a). You can use any letters, but they must be in the same format - a variable followed by another variable in parenthesis! Ok.. what does this really mean? Remember when we graphed linear equations? Every equation was written as y = ..... Well, now instead of y = , you are going to see f(x) ..... f(x) is another way of representing the "y" variable in an equation. Let's take a look at an example!

Notice y is replaced with f(x), g(x), even h(a). This is function notation. They all mean exactly the same thing! You graph all of these exactly as you would y = 2x +3. We are just using a different notation! In the next lesson, evaluating functions, you will see how much easier it is to use function notation!


Evaluating Functions

In our introduction to functions lesson, we related functions to a vending machine. You "input" money and your "output" is candy or chips! We're going to go back to that visual as we evaluate functions. We are going to "input" a number and our "output is the answer!

Let's take a look at a few examples!

Example 1

Did you notice how we just substituted for x and we found our answer? Not too hard, is it? Next you will see how using function notation makes it easier to display your answer if you are asked to evaluate the function more than one time!


Example 2

Do you see how easy it is to keep our answer organized since we have two answers to display? We label them as f(-2) and f(3) to keep them organized! I hope you are finding this to be pretty easy! You actually already know how to evaluate functions if you can evaluate equations. We are just giving it a different name!


Linear Functions

If you studied the writing equations unit, you learned how to write equations given two points and given slope and a point. If you did well with that unit, then studying linear functions will be a great review. If you had trouble with that unit, then you may want to go back and review the following concepts before starting this lesson:

• Writing Equations in Slope Intercept Form.

• Finding slope given two points.

• Writing equations given two points. Let's quickly review ordered pairs and function notation before moving on!

Ok.. now that you know how to write an ordered pair from function notation, let's look at an example!


Example 1

Ok, that was pretty easy, right? Yes...now do you see how Math has that spiral effect? You already knew this skill, but it's coming back in a different format! Next we are going to take it one step further and find the slope of the graph for a linear function. Take a look at this example.


Example 2

Now, let's take it one step further. We are going to find an equation for a linear function. Notice how we use steps from example 2 to start this problem.


Example 3


Solving a Linear Function - Part 2

In the previous lesson on functions you learned how to find the slope and write an equation when given a function. Click here to review the first lesson on Linear Functions before moving on! Linear functions are very much like linear equations, the only difference is you are using function notation (fx) instead of y. Otherwise, the process is the same. If you need a refresher on writing equations, you may want to review writing equations given slope and a point before completing this lesson. You may also need a refresher on solving equations to complete problem number 1.

Ok, let's move on! In our first example, we are going to find the value of x when given a function. Let's take a look.


Example 1

Pretty easy, right? This is really just a review of concepts that you've already learned! In example 2, you will see how to write the equation of a function given slope and a point.


Quadratic Functions - Lesson 1

Quadratic Functions? What in the world could they be? So far in our study of Algebra, we have discovered all of the ins and outs of linear equations and functions! We know that linear equations graph a straight line, so I wonder what a quadratic function is going to look like. Let's take a look! A quadratic function is always written as:

f(x) = ax2 +bx + c

Ok...let's take a look at the graph of a quadratic function, so that we can define a few vocabulary words that you will see associated with quadratic functions.

The graph of a quadratic function is called a parabola. A parabola contains a point called a vertex. The parabola can open up or down. If the parabola opens up, the vertex is the lowest point. This point is called the minimum point. If the parabola opens down, the vertex is the highest point. This point is called the maximum point. A parabola also contains two points called the zeros or some people call these the x-intercepts. The zeroes are the points were the parabola crosses the x-axis.


Now, we will use a table of values to graph a quadratic function!


Example 1


Notice that after graphing the function, you can identify the vertex as (3,-4) and the zeros as (1,0) and (5,0).


Using the Vertex Formula Quadratic Functions - Lesson 2

Before we begin this lesson on using the vertex formula, let's briefly recap what we learned in lesson 1. A quadratic function can be graphed using a table of values. The graph creates a parabola. The parabola contains specific points, the vertex, and up to two zeros or x-intercepts. The zeros are the points where the parabola crosses the x-axis. If the coefficient of the squared term is positive, the parabola opens up. The vertex of this parabola is called the minimum point.

If the coefficient of the squared term is negative, the parabola opens down. The vertex of this parabola is called the maximum point.


In the previous lesson, you graphed quadratic functions using a table of values. In that lesson, I gave you the x values within the table of values. How would you know which x values to choose if you were graphing a quadratic function on your own? How would you be sure that you chose an x value that allowed you to graph the vertex? This point is essential for graphing a parabola. There is a special formula that you can use to find the vertex. Once you know the vertex, you can be sure that you have the essential point for graphing the parabola!


The Vertex Formula

Now, let's look at an example where we use the vertex formula and a table of values to graph a function.


Example 1


The vertex, also known as your maximum point, is (-1, 4.5). The zeros of the function are: (-4,0) and (2,0). These are the points where the parabola crosses the x-axis.


Step Functions Also known as Discontinuous Functions

The graph below is an example of a step function. As you examine the graph, determine why you think it might be called a step function.

Do you see what looks like a set of steps? This is one reason why it is called a step function. It is better known as a discontinuous function. Why do you think it is called a discontinuous function? Yes, it is not a continuous line, it stops and starts repeatedly! So, the question may be, is it a function? Does it pass the vertical line test? Let's see!


It looks like the vertical lines may touch two points on the graph at the same time. However, take a look at the points. One is a closed circle and one is an open circle. If you review our inequalities lesson, you will remember that a closed circle means that the point includes that particular point. But... an open circle does NOT include that point. So, in this case, where it looks like the vertical line is touching two points, it is really only touching one point, because the open circle does not include that point. So, to answer our question, yes this is considered a function. It's not linear, and it's not quadratic. We call it a step function or a discontinuous function.

How Do We Read and Interpret a Discontinuous Graph? Let's take a look at our postage graph again.

This graph describes how much it will cost to send a letter depending on the weight of the letter. I've labeled the steps so that you better understand the explanation below. Step 1: If the weight of the letter is over 0 oz and up to 1 oz (including 1 oz, since the circle is closed), it will cost 39 cents. Step 2: If the weight of the letter is more than 1 oz (not 1 oz exactly because the circle is open) and up to 2 oz (including 2 oz since the circle is closed), then the price is 41 cents. Step 3: If the weight of the letter is more than 2 oz (not 2 oz exactly because the circle is open) and up to 3 oz (including 3 oz since the circle is closed), then the price is 43 cents.


Steps 4-6 follow the same pattern as steps 1-3 described above. As you can see, this graph tells you exactly how much your letter will cost depending on the weight. A discontinuous graph must be used because the price stays the same between ounces, but then changes to the next price as you reach a whole ounce. Let's take a look at a few other discontinuous graphs and determine whether or not they are functions. These graphs may not look like "steps", but they are considered discontinuous.

This graph is not a function because when utilizing the vertical line test, it touches in two points. Both points at x = 1 are solid, therefore the graph is discontinuous, but not a function.

This graph is a function because it passes the vertical line test. Each vertical line only touches the graph at one point. (Although it looks like it touches at two points at x = -3, since one circle is "open" we do not include that as a point.) Therefore, it is considered a discontinuous function. It is discontinuous at x = -3.

Let's practice creating and interpreting a graph for a discontinuous function.


Example 1

A wholesale t-shirt manufacturer charges the following prices for t-shirt orders:

• $20 per shirt for shirt orders up to 20 shirts.

• $15 per shirt for shirt between 21 and 40 shirts.

• $10 per shirt for shirt orders between 41 and 80 shirts.

• $5 per shirt for shirt orders over 80 shirts.

• Sketch a graph of this discontinuous function.

• You've ordered 40 shirts and must pay shipping fees of $10. How much is your total order?

Solution

If I ordered 40 shirts and must pay $10 in shipping fees, then my total order will cost $610. (40 * $15) +10 = 610.


Understanding Exponents

As we begin our study o f monomials, you will need to learn and understand the use of exponents. So, let's begin by defining the term exponent. An exponent is a number (small and raised) that represents the "shortcut method" to showing how many times a number is multiplied by itself. That sounds complicated, so let's look at a few examples:

Example 1

Example 2

Your base can even be a negative number! Take a look!


Example 3

Example 4


Tip!

Whenever you have a negative base and the exponent is even, your answer will always be positive! Whenever you have a negative base and the exponent is odd, your answer will always be negative! Now is the tricky problem! What happens when you have a negative base, but it's not in parenthesis?

Example 5

Now, I have just one more tip for you when working with exponents!

Tip!

When you have a 0 as an exponent, your answer will always be 1. The only exception is 00 is undefined.

Examples

40 = 1 or 80 = 1


Using the Laws of Exponents

Before you begin working with monomials and polynomials, you will need to understand the laws of exponents. There are three laws or properties that I am going to discuss in this lesson. We will look at the following properties:

• Multiplying Powers with the Same Base

• Power of a Power Property

• Power of a Product Property For each of the three laws, we will write a few examples in expanded form. This will help you to understand why the law works. Then I will define the property. Finally we will look at a few examples.


Multiplying Powers with the Same Base

Tip

When a term does not contain an exponent, it is assumed to be 1. For example:

3 = 31 y = y1 r = r1


Power of a Power Property

Power of a Product Property

Before we start this one, let's define the word product. A product is the answer to a multiplication problem. So, power of a product means that we are raising a multiplication problem to a power. Take a look at this example:


What is a Monomial?

Before we get into all the fun things we can do with monomials, we better first define a monomial! It sounds like a strange word, but let's look at its prefix! Monomial - the prefix mono means one A monomial is one term! A monomial can be any of the following:

In the equations unit, we said that terms were separated by a plus sign or a minus sign! Therefore:

Tip

A monomial CANNOT contain a plus sign (+) or a minus (-) sign!

Let's take a look at one more example!


Of the following three terms, which one is NOT a monomial? 1. 9xyz 2. (3xy)2 3. 2x+2y3

Solution:

Did you pick number 3? If so, you are correct! 2x+2y3 is not a monomial because there is a plus sign. In this case, we are adding 2 monomials!


Multiplying Monomials

Before you begin reading this lesson on Multiplying Monomials, please make sure that you've read the lesson on laws of exponents and monomials. These two lessons will definitely be needed in order to understand how to multiply monomials!

When you multiply monomials, you will need to perform two steps:

• Multiply the coefficients (constants)

• Multiply the variables. Let's look at a few examples.

Each line of these examples shows a different step. I broke it down step by step for you to see the exact process. You will be able to do a lot of this work mentally, which I will show you later on. I wanted you to understand each step, that's why these explanations are so long!

Example 1


Example 2

One more example, but this time I'm not going to show each individual step. As you master this skill, this is the way in which you will multiply monomials! It's actually pretty quick!


Example 3


Dividing Monomials How Do We Divide When Exponents are Involved?

As you've seen in the prior lessons, when we work with monomials, we see a lot of exponents. You've discovered the laws of exponents and the properties for multiplying exponents, but what happens when we divide? That is the question we are going to answer in this lesson! Let's start by taking a look at a few problems in "expanded form". Once you examine these examples, you'll discover the rule on your own!

Expanded Form Examples

Take a look at the exponents in the original problem and then analyze the exponents in the answer for each example. Can you figure out the rule for exponents when you are dividing?

When you divide powers that have the same base, you subtract the exponents. That's a pretty easy rule to remember! It's the opposite of the multiplication rule! When you multiply powers that have the same base, you add the exponents and when you divide powers that have the same base, you subtract the exponents! Let's look at a couple of examples!


Example 1

Example 2

That's a pretty easy rule to remember! Let's take a look at one more property. This property is called, Power of a Quotient Property. So, what is a quotient? A Quotient is an answer to a division problem. Let's take a look at what happens when you raise a fraction (or a division problem) to a power. Remember: A division bar and fraction bar are synonymous!

Power of a Quotient Property

To find the power of a quotient, raise the numerator to the power, and the denominator to the power. Then divide! Let's take a look at a few examples!


Power of a Quotient Example 1

Power of a Quotient Example 2


Simplifying Monomials

Now I want you to think about everything you've learned about the laws of exponents, multiplying monomials, and dividing monomials. When simplifying monomials, you will need to use ALL of this information! This is fun, let's get started by looking at a quick summary of the properties of exponents that are described in the lessons above!

Properties of Exponents and Using the Order of Operations

• If you have a combination of monomial expressions contained within grouping symbols (parenthesis or brackets), these should be evaluated first!

• Power of a Power Property - (This is similar to evaluating Exponents in the Order of Operations). Always evaluate a power of a power before moving on the problem!

Example of Power of a Power

• When you multiply monomial expressions, add the exponents of like bases.

Example of Multiplying Monomials

\

• When you divide monomial expressions, subtract the exponents of like bases.


Example of Dividing Monomials

Ok... that was just a quick review. If you need more of a review, please go back and review the entire lesson! Now let's look at a couple of examples!

Example 1


Example 2


Negative Exponents and Zero Exponents

So far in this unit, you've learned how to simplify monomial expressions with positive exponents. Now we are going to study two more aspects of monomials: those that have negative exponents and those that have zero as an exponent. I am going to let you investigate to see if you can come up with the rule on your own! Take a look at the following problems and see if you can determine the pattern.

Can you figure out the rule? If not, here it is...

The Rule for Negative Exponents:

The expression a-n is the reciprocal of an

TIP:

A reciprocal is when you "flip a fraction".

Examples: The reciprocal of 3/4 is 4/3.

The reciprocal of 5 is 1/5. (You can make a whole number a fraction by putting a one in the

denominator: 5 = 5/1)

***An easy rule to remember is: if the number is in the numerator (top), move it to the denominator (bottom). If the number is in the denominator, move it to the numerator!

Examples


Now let's quickly take a look at monomials that contain the exponent 0. Any number (except 0) to the zero power is equal to 1.

Not too hard, is it? Let's look at a couple of example problems and then you can practice a few!

Example 1

Example 2

**Since 2/3 is in parenthesis, we must apply the power of a quotient property and raise both the 2 and 3 to the negative 2 power. First take the reciprocal to get rid of the negative exponent. Then raise (3/2) to the second power.


Example 3


Scientific Notation

Since we've done so much work with exponents, we are going to continue our studies by looking at scientific notation. Scientific notation is simply a "shorter way" of writing very large or very small numbers. Scientists often work with very large or extremely small numbers when performing experiments, which is why it is called "scientific" notation. There is a specific format in which you must write a number in order for it to be considered scientific notation. Let's take a look:

It's really important that you understand the format for scientific notation, so let's look at a chart that illustrates a few incorrect and correct ways to write numerals in scientific notation.


Are you wondering why we have to multiply by 10 to a power? Why 10 and not 5 or 3? Take a look at this quick example.

Do you see the pattern? We continue to add a 0, or another place to the number three. Notice that the three itself remains unchanged. We just continue to add a 0, when we multiply by a larger power of 10.

Shortcut:

When you multiply by a positive power of 10, you can simply move the decimal point to the right the same amount of spaces as the exponent.


Let's take a look at a couple of examples of numbers written in scientific notation and rewrite them in expanded form.

Example 1


Example 2

Now, let's look at a few examples in expanded form and rewrite them in scientific notation.


Example 3


Example 4

So, how are you feeling about scientific notation when the power of 10 is positive? It gets easier as you practice a little more, I promise! Are you ready to look at a few problems where the power of 10 is negative? Don't worry - same process, we're just going to move the decimal in a different direction! If you studied the lesson on negative exponents, then you learned that when you raise a whole number to a negative power, it results in a fraction because you take the reciprocal. And... we all know that proper fractions have a smaller value than whole numbers. So, when we work with powers of 10 to a negative power, we are going to be working with very small numbers!


TIP:

If you have a negative power of 10, you must move the decimal point to the left! We are creating a smaller number!

Let's take a look at a couple of examples written in scientific notation and expand them!

Example 5


Example 6

Let's look at one more example! I've given you the expanded form this time, and would like to show you how to write it in scientific notation. Pay special attention to where I stop counting when I move my decimal to the right!


Example 7


Using Scientific Notation When Multiplying and Dividing Monomials

In order to understand this lesson, you will need to have a basic understanding of scientific notation.

Click here to review the lessons on multiplying and dividing monomials, which may also assist you in completing this lesson!

Throughout your studies, you may be asked to multiply or divide numerals that are written in scientific notation. You will follow the same process that you learned when multiplying and dividing monomials; however, the only difference is that you will need to make sure that your final answer is written in scientific notation.

Let's look at a few examples.


Multiplying Monomials Written in Scientific Notation

Example 1


Example 2

Now let's look at a couple of examples that require division!


Dividing Monomials Written in Scientific Notation

Example 3


Example 4

It can be tricky trying to figure out the exponent if the final answer is not in scientific notation format. If you have trouble thinking it through, write the initial answer in expanded form and then translate it back into scientific notation. Take a look at this final example.


Example 5

Obviously this is a much longer process, so you only need to do this if you are really confused about how to convert the final answer.

Once you do this a few times, you'll probably be able to convert to the shorter method - it will suddenly make sense!


Introduction to Polynomials

So, what is a Polynomial? This is definitely not a word that we hear every day!

A Polynomial is a finite sum of terms. This includes subtraction as well, since subtraction can be written in terms of addition. Let's take a look at a couple of examples and this will make more sense!

Examples of Polynomials

2x2 + 3x - 5

2x2y2 + 3xy - 5xy2

5x + 3y +6x +2y

As you can see from the examples above, we are simply adding (or subtracting) two or more terms together!

Polynomials can also be classified according to the number of terms. Let's take a look!

REMEMBER: Terms are separated by a plus sign or a minus sign!

Let's take a look at one more definition! The degree of a polynomial with one variable is the highest power to which the variable is raised. Take a look!


Degree of Polynomials

Polynomials in one variable should be written in order of decreasing powers. If this is the case, the first term is called the leading coefficient. The exponent of this first term defines the degree of the polynomial.

Now, for one last definition, which is actually a review! If two or more terms have exactly the same variables, then they are called like terms!

Like Terms


Adding Polynomials

Before we begin adding polynomials, we are going to review the definition of like terms.

Like terms are two or more terms that have identical variables, and possibly different coefficients.

Take a look at the following examples:

It's very important that you understand this definition of like terms as you begin working with polynomials!

When you add polynomials, you are simply going to add the like terms! There are two methods that you can use to add polynomials: the vertical method or horizontal method. I will show you both methods, so that you can choose the one that is most comfortable for you!

For example 1, I will use the horizontal method.

Example 1

Did you notice how we simply rewrote the problem with like terms together, and then combined the like terms? Now, I'll show you the vertical method.


Example 2

Let's take another look at an example using the horizontal method.

Example 3

Now we'll look at an application of this skill and use the vertical method to solve!


Example 4


Subtracting Polynomials

Subtracting Polynomials is very similar to adding polynomials! In fact, we will be changing the subtraction problem to an addition problem!

In the Pre-Algebra section of the website, we started out by reviewing integers.

We said, "When you subtract integers, you must add the opposite! We also talked about the Keep - Change- Change Rule! That rule applies to polynomials as well!

Take a look at these examples that show you how to rewrite the problem as an addition problem.

Your first step is to change the subtraction problem to an addition problem. Then you add, just as you did in the adding polynomials lesson!

Let's take a look at an example! Once we change this problem to an addition problem, we will use the horizontal method for solving!


Example 1

For our next example, I am going to add a set of brackets, which will require a few more steps in order to reach a solution! Watch carefully!


Example 2

One last example! This example will demonstrate how to use the vertical method.


Example 3


Multiplying Polynomials

Multiplying Polynomials requires two prerequisite skills! You must be able to apply the laws of exponents and the distributive property. The main rule that you need to remember is:

When you multiply terms that contain powers:

Multiply the coefficients and add the exponents!

Ok.. now let's move on!

In our first example, pay close attention to how the distributive property is used.


Example 1

Did notice the distributive property? As we distributed the term, we multiplied the coefficients and added the exponents of like bases. That was a pretty easy example!

Let's look at a problem that's a little more complex.

Example 2

This skill isn't too hard if you know the laws of exponents and know how to use the distributive property.


Multiplying Binomials Using the Foil Method

Multiplying polynomials becomes a little trickier when you multiply two binomials! We are still going to use the distributive property, but many students refer to the acronym, FOIL in order to remember the steps for multiplying binomials!

I am actually going to show you two ways to multiply binomials! The first way is thinking of it as another way to use the distributive property. The second way will be to use the FOIL Method. Let's take a look at the following examples.

Example 1 Using the Distributive Property

We will use the distributive property to multiply the following binomials:

(3x - 4)(2x +1)

Now, that seems like a lot of work doesn't it? That is the process for multiplying binomials, but eventually you'll be able to complete this process without writing out each step of the distributive property. I wrote it out, in hopes that you will understand the process. You will be able to do a lot of those steps mentally!


One way to help you remember the steps to perform mentally is to remember the acronym, FOIL. If you compare each step in the first example, to the steps used in example 2 with the foil method, you will find that they are pretty much the same! Let's take a look!

FOIL stands for:

Example 2


As you can see, when using FOIL ,we first distributed the 3x throughout the quantity (2x +1). Then we distributed the -4 throughout the quantity (2x + 1). So, you can think of it as distributing one binomial throughout the other or you can remember FOIL to perform the same steps!

Let's look at another example using FOIL!

Example 3 Using the FOIL Method

TIP

There is one thing that you need to remember with FOIL! It only works when you are multiplying two binomials! This is not the only method used when multiplying polynomials and it doesn't work for all polynomials! It ONLY works with two binomials!


The Square of a Binomial

In this lesson, we will discover a special rule that can be applied when you square a

binomial.

In our last method, we studied the FOIL method for multiplying binomials. We can still apply the FOIL method when we square binomials, but we will also discover a special rule that can be applied to make this process easier. Let's take a look at Example 1.

Example 1


Let's take a look at a special rule that will allow us to find the product without using the FOIL method.

The square of a binomial is the sum of: the square of the first terms, twice the product of the two terms, and the square of the last term.

I know this sounds confusing, so take a look..

Now let's take a look at Example 1 and find the product using our special rule.

Example 1 Using the Special Rule

Now let's take a look at another example. This time we are going to square a binomial, but this binomial will contain a subtraction sign.


Example 2

For this example, we will not use FOIL, we will use our special rule!

Let's quickly recap, and look at the definition for Squaring a Binomial. You might want to record this in your Algebra notes.

Squaring a Binomial


Multiplying Binomials - A Special Case Products That Result in the Difference of Two Squares

Another frequently occurring problem in Algebra is multiplying two binomials that differ only in the sign between their terms. An example would be:

(x-4)(x+4)

Notice that the only difference in the two binomials is the addition/subtraction sign between the terms.

We will solve this problem using the FOIL in Example 1. Then we will look at a special rule that can be applied to make this problem much easier to multiply.

Example 1 Using the FOIL Method


Did you notice how the middle terms added up to 0? This will happen every time you multiply two binomials whose only difference is the sign between the terms (+ and -).

The rule for multiplying this kind of binomial is:

When multiplying binomials whose only difference is the sign between the two terms, square the first term, square the second term, and subtract.

Let's take a look at the first example and apply this new rule.

Example 1 Using our Special Rule

The expression x2 - y2 is called the difference of two squares.

Let's take a look at one more example using our special rule.


Example 2

Yes, I know what you are thinking... it is much easier to use the special rule.

However, you need to remember that this is a "special case" and this rule ONLY works when the binomials only differ by the plus and minus sign between the terms.


Factoring Polynomials

Lesson 1 - Using the Greatest Common Factor (GCF)

There are several methods that can be used when factoring polynomials. The method that you choose, depends on the make-up of the polynomial that you are factoring.

In this lesson we will study polynomials that can be factored using the Greatest Common Factor.

Make sure that you pay careful attention not only to the process used for factoring, but also to the make-up of the polynomials that can be factored using this method.

Let's start by looking at the definition of factors.

When you factor a polynomial, you are trying to find the quantities that you multiply together in order to create the polynomial.

Take a look at the following diagram:

Now let's talk about the term greatest common factor.

The greatest common factor (GCF)for a polynomial is the largest monomial that is a factor of (divides) each term of the polynomial.

Note: The GCF must be a factor of EVERY term in the polynomial.


Take a look at the following diagram:

Before we get started, it may be helpful for you to review the Dividing Monomials lesson. You will need to divide monomials in order to factor polynomials.

Let's take a look at a couple of examples.


Example 1


Now, let's take a look at an example that involves more than one variable!


Example 2

And... one last example.


Factoring in Algebra

Lesson 2 - Factoring by Grouping

Factoring in Algebra can be accomplished in many different ways. When it comes to polynomials, each situation is different based on the make-up of the polynomial. In our last lesson, we learned how to factor by using the greatest common factor.

However, some polynomials have no greatest common factor other than 1. Therefore, we would need to choose another method for factoring.

In this case, we would look to see if the polynomial has a couple of terms with a common factor. If so, we can group them together and factor separately.

Take a look at the following example:

Example 1

3x2 - 3 + x2y - y

There are 4 terms in the polynomial. However, there are no common factors within the 4 terms.

Do you see two terms that have a common factor that could be grouped together?


I know that factoring can be confusing, but think of factoring as rewriting the problem using the distributive property. You want to continue factoring a polynomial until no common factors exist.

Let's look at another example.


Example 2


Factoring Trinomials – Lesson 3

Factoring trinomials is probably the most common type of factoring in Algebra. In Algebra 1 we will factor trinomials that have a lead coefficient of 1. In Algebra 2, we will progress to factoring more complex trinomials whose lead coefficient is greater than 1.

To begin this lesson, it is important for you to understand the process of multiplying binomials using the FOIL method. Please be sure to review that lesson before starting this lesson. Click here to review the FOIL method lesson.

The diagram below outlines the product of multiplying two binomials. It's important to understand how we reach the trinomial because in this lesson we are going to work backwards to form the factors or two binomials.

Did you notice how we added the two last terms of each binomial (3 & 5) to get the middle term and we multiplied the same two last terms (3 & 5) in order to get the last term of the trinomial?


Example 2


TIP

When have a trinomial with a minus sign, pay careful attention to your positive and negative numbers. In the example above, 8 and -2 are the numbers that we needed to complete our binomials; however, -8 and 2 would not have worked!

I know that factoring trinomials is tough, so let's look at one more example. Again, this trinomial will contain a minus sign, so pay careful attention to the positive and negative numbers that you choose.


Example 3


Solving a Quadratic Equation

This unit is an introductory unit to Quadratic Equations. We will study square roots, the Pythagorean Theorem and solving simple quadratic equations using a variety of methods.

This unit is designed to teach you the basic principles for Quadratic Equations, which will allow you to be more successful in Algebra 2.

What is a Quadratic Equation?

A quadratic equation is any equation that can be written in the form:

Which of the Following Equations Are Quadratic?


Introduction to Square Roots

In the monomials unit, we took a close look at exponents and powers. You may want to review the lesson on exponents before starting this lesson on square roots.

Let's get started.

Another term for raising a number to the 2nd power is "squaring a number". For example:

22 = 4. This can be read as 2 "squared" equals 4. This means that 2 x 2 = 4.



As you've probably discovered in Math, there is always an "opposite" operation! So, can you guess the opposite operation for "squaring" a number?

You've got it - taking the "square root" is the opposite of squaring a number!

Let's take a look at the symbols.

Before we dig in, let's look at some vocabulary that you should be familiar with. The symbol used for identifying roots is called the radical sign.

The number inside the radical sign is called the radicand.

When you start Algebra 2, you will also learn that if you are working with cube roots, or fourth roots..., there will be another number called the index. We'll worry about the index later.

Take a look at the diagram below to further explain these definitions.


Take a look at the chart below for a list of the most common square roots.

Tip

It is a common mistake to identify the square root of 4 as 2 and -2.

Why did I stress that if you want a negative square root, that the negative sign must be outside of the radical sign?


Wow! We've covered a lot in this lesson, but let's look at one more example.

What Happens When You Take the Square Root of a Fraction?

When you take the square root of a fraction, simply take the square root of the numerator and the square root of the denominator. The answer remains a fraction. Take a look at the following examples.

Wow! We've accomplished a lot, and there's a lot more to learn. But, we'll take it one step at a time, and soon you'll have a solid understanding of quadratics.

Now, you are ready to start our first lesson on solving quadratic equations.


How to Solve Quadratic Equations

These equations do look intimidating and more difficult to solve. But... don't worry, I am going to show you several methods for solving quadratic equations.

You can solve a simple quadratic equation using the same rules that you used when solving linear equations. We'll discover that method in this lesson.

You can also solve quadratic equations by using the Quadratic Formula and by Factoring.

For right now, let's focus our attention on solving simple quadratic equations using a few strategies that you are already familiar with.

There is, of course, one new skill that you must apply. That is using square roots. Knowing how to take the square root of a number is essential to this lesson, so please review this lesson if needed.

Let's look at our first example, which is an extremely basic equation. Take note of the use of the square root.

Example 1


For Example 2, we will look at a two-step quadratic equation. In this example we will get rid of the constant first and then take the square root of x2 in order to get x by itself. Take a look.

Example 2

Now we'll look at an example that it a little more complex because it requires one more step in order to get rid of the coefficient of x2.


Example 3

Example 4


Pythagorean Theorem

This man, is Pythagoras. He was one of the first Greek Mathematicians. He discovered an amazing property of the right triangle that we all now know and love as the Pythagorean Theorem.

This theorem is well known in Algebra and we study it in our Quadratics unit because we are working with squared terms. Take a look.

The Pythagorean Theorem

In any right triangle, the sum of the squares of the legs (2 shorter sides) is equal to the square of the hypotenuse (the longest side).

Please Note: This theorem ONLY works for Right Triangles.


Let's start by taking a look at an example where we need to find the hypotenuse.

Example 1

In a right triangle, the length of one leg is 6 cm and the length of the other leg is the square root of 13 cm. Find the length of the hypotenuse.

Now let's look at an example of a problem where we are asked to find the length of a leg of a right triangle.


Example 2

A ladder that is 16.5 feet tall is placed against the side of a tree. The base of the tree to the top of the ladder is a distance of 14 feet. How far is the ladder placed away from the base of the tree?

If word problems confuse you, the best thing you can do is draw a picture!


Factoring Quadratic Equations

Are You Ready to Use Your Knowledge of Factoring to Solve Equations?

In my polynomials unit, I demonstrated how to factor trinomials. But, did you ever wonder WHY we need to factor trinomials?

Well, now you are about to find out!

By factoring quadratic equations, we will be able to solve the equation. There are several different ways to solve a quadratic equation. If the equation can be factored, then this method is a quick and easy way to arrive at the solution.

It is EXTREMELY important that you understand how to factor trinomials in order to complete this lesson. If you do not have a thorough understanding of factoring quadratic equations, please review this lesson now!

Let's get started!

We can factor quadratic equations in order to find a solution to the equation because of the Zero-Factor Property.

The Zero-Factor Property

For all real numbers, x and y:

x(y) = 0

if and only if x = 0 or y = 0 (or both)

I know math properties and theorems can be difficult to understand at times. So, the zero-factor property simply means that if the equation is equal to 0, then at least one of the factors must be equal to 0.

Think about it: When you multiply two or more numbers, the only way to get an answer of 0, is if one of the numbers that you are multiplying is 0. That same, very basic property for multiplication applies here.

The key here is that the equation must be set equal to 0.

Let's take a look at an example.


Example 1

Pretty easy, if you know how to factor trinomials! Let's take a look at another example.


Example 2


Quadratic Formula

When in doubt, use the quadratic formula! This formula allows us to solve any type of quadratic equation.

If the equation can be factored, factoring is usually the quicker method. However, not all equations can be factored easily. In this case, the quadratic formula always works. It just takes a little more time due to the heavy computation. But... it's not hard!

The Quadratic Formula

Once you substitute the values for a, b, and c, you'll just need to use some basic math to arrive at the solution. It's really pretty easy. Take a look!


Let's take a look at one more example. We'll look at a real world application using quadratic equations.

Example 2


Graphing Quadratic Equations

I first introduced the concept of graphing quadratic equations in our Functions unit. In this unit, we discovered how to use a table of values in order to graph a quadratic function. This would be a great lesson to review, as you will see a lot of vocabulary that relates to graphing parabolas.

And... in case you didn't remember, the graph of a quadratic equation is called a parabola. In the following examples, we will pull together all of our knowledge for quadratic equations to create the graph.

Before beginning this lesson, please make sure that you fully understand the vertex formula, factoring quadratic equations, and the quadratic formula.

One way to graph a quadratic equation, is to use a table of values. While this method works for every quadratic equation, there are other methods that are faster.


For any quadratic equation in the form: y = ax2 + bx + c

The graph will result in a parabola.


So, how do we find all of these points in order to create the graph?

We need to find the vertex, x intercepts, and y intercept. So, let's look at an example and I will show you how to find all the points needed!

Example 1


Wow! That seems like a lot of work doesn't it? It's really not too bad once you do it a few times.

Did you notice how some of the information that you learned in previous chapters is coming up again? For example, we talked about the vertex formula when we graphed quadratic functions. We also talked about x-intercepts and y-intercepts when we graphed linear equations.

When we graphed linear equations, we let y = 0 when we were trying to find the x-intercepts and we let x = 0 when we were trying to find the y-intercept. We just used the same process for quadratic equations.

This is what I Love about Algebra! You really never forget the concepts because you use them over and over again.

Ok... are you ready to look at one more example? We will graph a quadratic equation that you may not be able to factor as easily. In order to find the x intercepts, we will use the quadratic formula instead of factoring.


Math Probability - What a Fun Unit!

In math, probability is the likelihood that an event will happen. It is the ratio of the number of ways an event can occur to the number of possible outcomes.

Probability is expressed as a fraction or decimal from 0 to 1. Think of the following scale when determining the probability of an event occurring:

A probability of 1 means that you are absolutely certain that an event will occur. For example, if you have a coin, the probability of flipping the coin and it landing on heads or tails is 1. Unless someone has a trick coin, you can be certain that either a heads or tails will show when flipped.

If you flip the same coin, you could also say that the probability of flipping heads is .5 or 1/2. You have 1 chance out of 2 to flip heads. Therefore, you have a 50/50 chance.

If something has a probability of 0, then that means that it is impossible, or there is no chance of the event occurring.

Then, there's all of the other fractions (and decimals) between 0 and 1 that are used to represent probability. As we explore this unit on probability, you will discover that you will always write probability as a fraction.

For most courses in middle or high school math probability is a popular unit of study. Why? Without realizing it, you use probability every day.

What is the probability that I will get the job? What is the probability that I will be able to win the contest? What is the probability that I will get the red gumball out of the machine? Yes.... you may not have been actually computing the probability when you were 6, but I bet that you were analyzing the probability of getting that gumball or "prize" out of the candy machine.


Use These Examples of Probability To Guide You Through Calculating the Probability of Simple Events.

Probability is the chance or likelihood that an event will happen. It is the ratio of the number of ways an event can occur to the number of possible outcomes. We'll use the following model to help calculate the probability of simple events.

As you can see, with this formula, we will write the probability of an event as a fraction. The numerator (in red) is the number of chances and the denominator (in blue) is the set of all possible outcomes. This is also known as the sample space.

Let's take a look at a few examples of probability.


Example 1

Now let's take a look at a probability situation that involves marbles.


Example 2

Hopefully these two examples have helped you to apply the formula in order to calculate the probability for any simple event.


Using the Fundamental Counting Principle to Determine the Sample Space

As we dive deeper into more complex probability problems, you may start wondering, "How can I figure out the total number of outcomes, also known as the sample space?"

We will use a formula known as the fundamental counting principle to easily determine the total outcomes for a given problem. First we are going to take a look at how the fundamental counting principle was derived, by drawing a tree diagram.

Example 1

We were able to determine the total number of possible outcomes (18) by drawing a tree diagram. However, this technique can be very time consuming.


The fundamental counting principle will allow us to take the same information and find the total outcomes using a simple calculation. Take a look.

Example 1 (continued)

As you can see, this is a much faster and more efficient way of determining the total outcomes for a situation.


Example 2

I would not want to draw a tree diagram for Example 2! However, we were able to determine the total outcomes by using the fundamental counting principle.

Let's look at one more example and see how probability comes into play.


Example 3


Your Source for Probability Problems & Examples of Independent Events

In this lesson, we will determine the probability of two events that are independent of one another. Let's first discuss what the term independent means in terms of probability.

Two events, A and B, are independent if the outcome of A does not affect the outcome of B.

In many cases, you will see the term, "With replacement". As we study a few probability problems, I will explain how "replacement" allows the events to be independent of each other.



Example 1

Example 1 is pretty easy to comprehend because we are finding the probability of two different events using two different tools. Let's see what happens when we use one tool, like a jar of marbles.


Example 2


Are You In Need of Probability Help?

This lesson provides a thorough study of Probability with Dependent Events

Have you been searching for probability help, specifically with dependent events? If you are new to Algebra-class.com or just starting a probability unit, you may want to take a look at the introductory probability lesson or the lesson on independent events. But... if you are ready to study dependent events, let's take a look at the definition.

Dependent Events

Two events, A and B, are dependent if the outcome of the first event does affect the outcome of the second event.

In many cases, the term "without replacement" will be used to signify dependent events.

Dependent Events are notated as:

P(A, then B)

Let's take a closer look at situations with dependent events.


Example 1

Did you notice how the playing card was not replaced, so the outcomes and sample space were reduced for the second event? The second event is dependent on what happens on the first pick. Since this is theoretical probability and we don't know what would really happen on the first pick, we always assume that the first event happens as stated in the problem.



Example 2


Theoretical Probability versus Experimental Probability

You've heard the terms, theoretical probability and experimental probability, but what do they mean? Are they in anyway related? This is what we are going to discover in this lesson.

If you've completed the lessons on independent and dependent probability, then you've already found the theoretical probability for numerous problems.

Theoretical Probability

Theoretical probability is the probability that is calculated using math formulas. This is the probability based on math theory.

Experimental Probability

Experimental probability is calculated when the actual situation or problem is performed as an experiment. In this case, you would perform the experiment, and use the actual results to determine the probability.

In order to accurately perform an experiment, you must:

• Identify what constitutes a "trial".

• Perform a minimum of 25 trials

• Set up an organizer (table or chart) to record your data.

Let's take a look at an example where we first calculate the theoretical probability, and then perform the experiment to determine the experimental probability. It will be interesting to compare the theoretical probability and the experimental probability. Do you think the two calculations will be close?

Example 1

This problem is from Example 1 in the independent events lesson. We calculated the theoretical probability to be 1/12 or 8.3%. Take a look:


Let's take a look at another example where the experimental probability may not be so easy to set-up.


I hope this lesson helps you to distinguish between theoretical and experimental probability. Experimental probability can actually be a lot fun!


Probability Lessons

This lesson focuses on Probability of Compound Events

Thus far, we've studied several probability lessons. If you want to review a few of these lessons before studying compound events, check out the lessons on the fundamental counting principle, independent events, and dependent events.

If you are ready, let's move onto finding the probability of compound events.

Compound events can be further classified as mutually exclusive or mutually inclusive. The probability is calculated differently for each, so let's first take a look at mutually exclusive events.

Compound Events That Are Mutually Exclusive

When two events cannot happen at the same time, they are mutually exclusive events.

For example, you have a die and you are asked to find the probability of rolling a 1 or a 2. You know when you roll the die, only one of those numbers can appear, not both. Therefore, these events are mutually exclusive of each other.

Mutually Exclusive Events (Events that cannot happen at the same time)

P(A or B) = P(A) + P(B)

Take note: With this formula, you are adding the probabilities of each event, not multiplying.

Let's take a look at an example of mutually exclusive events.


Example 1


Mutually exclusive events are pretty straightforward. Now let's take a look at compound events that are inclusive.

When two events can occur at the same time, they are inclusive.

For example, let's take our example of rolling a regular 6-sided die. You are asked to find the probability of rolling a 2 or an even number. These events are inclusive because they can happen at the same time. A 2 is an even number, so this would satisfy both, but you could also roll a 4 or 6. Because 2 is a even number, these are inclusive events.

Inclusive Events (Events that CAN happen at the same time)

P(A or B) = P(A) + P(B) - P(A and B)

This is a little trickier, so let's take a look at example of inclusive events.


Example 2


I know that compound events can be confusing, but first you must determine if the events are exclusive or inclusive.

If the events are exclusive, then just add the probabilities of each individual event.

If the events are inclusive, you must remember to subtract the number outcomes that occur in both events.


Odds and Probability

Identifying the odds of something happening is a little different that calculating the probability. It is written as a ratio; however, it is not written as a fraction.

Odds

The odds in favor of an event is the ratio of the number of ways the outcome can occur to the number of ways the outcome cannot occur.

# of ways the event CAN occur : # of ways the event CANNOT occur.

This is actually a lot easier than probability. So, let's take a look at an example.


Example 1

Odds and probability is pretty easy! Just remember to use a colon instead of a fraction. Also, remember that you are comparing the number of ways the outcome can occur to the number of ways the outcome cannot occur (not the total outcomes).


Geometric Probability

Although you may have never formally heard the term "geometric probability", I bet you've often thought about it. Have you ever played darts? If so, have you thought about what your chances are of landing the dart on the bulls eye? If so, you were thinking about geometric probability.

At first, geometric probability looks difficult. But, if you keep in mind the formula for basic probability, you will have no problems.

Formula for Probability

# of favorable outcomes / # of total outcomes



Example 1


Example 2


Geometric probability is kind of fun, isn't it? It's brings out the kid in you when you start thinking of dart games...

This completes our unit on Probability.

I hope that all of the explanations and examples in this e-book have helped you with your Algebra studies. If you have any questions, please feel free to contact me. Wishing you the best, Karin Hutchinson

simple steps to success in algebra

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