2.1 simplifying algebraic expressions - blinn college 0310 notes-exam_1… · 2.1 simplifying...

2.1 Simplifying Algebraic Expressions

A term is a number or the product of a number and variables raised to powers. The

numerical coefficient of a term is the numerical factor. The numerical coefficient of 3x

is 3.

Example 1

Identify the numerical coefficient of each term:

a) 9x

b) 3y−

c) x−

d) 22.7x y

__________________ contain the same variables raised to the same powers. Terms

that are not like terms are called unlike terms.

Like Terms Unlike Terms

3x, 2x 5x, 5x2

Example 2

Indicate whether the terms in each list are like or unlike:

a) 6x , 3x−

b) 2xy− , 2x y−

c) 5ab , 1

2ba−

d) 3 22x yz , 3 3x yz−

Eby, MATH 0310 Spring 2017

Simplifying the sum or difference of like terms is called__________________________. To combine like terms add or subtract the numerical coefficient and leave the variable part the SAME!

Example 3

Simplify each expression by combining any like terms:

a) 7 2 4x x− +

b) 9 2 1 6 7y y− + − + + −

c) 2 2 2 23 5 2x y xy x y xy+ − +

Sometimes you will first have to simplify using the ______________________ before

you can combine like terms. To use the distributive property multiply each term inside

the parentheses by what is in front of them. Be careful with negatives!

Example 4

Simplify each expression. Use the distributive property to remove any parentheses.

a) ( 5 6 2 )m n p− − + −

b) 1

(6 9)3x−


c) 14(2 6) 4x + −

d) 3(2 5) ( 7)x x− − +

And then there will be times that you need to translate from word to algebraic

expressions. In order to do so you might need to know what some words mean:

+ : sum, added, more than, increased by, plus, total of, together

- : subtracted from, less than, difference between, decreased, minus

x : product, times, any ⅜(etc.) of or % of, multiplied by

÷ : divided, quotient, ratio, over, per

Example 5

a) Triple a number, decreased by six.

b) Six times the sum of a number and two, increased by three.


2.2 The Addition and Multiplication Properties of Equality

Last time we talked about simplifying expressions, this time we will solve linear

equations. Before we do that let’s look at the difference between expressions and

equations!

Expressions Equations

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

ax b c+ =

where a, b, and c are real numbers and 0a ≠ .

In order to solve linear equations we will need to know some properties!

The ______________________________ states that if a, b, and c are numbers and if

a b= , then a c b c+ = + . This is also true for subtraction! So if a b= , then ca c b= .

Example 1

Solve each equation:

a) 6 18y − = b) 18 5x− = +

c) 2 3

3 4a + = −


The ______________________________ states that if a, b, and c are numbers and if

a b= , then a c b c• = • . This is also true for division! So if a b= , then ca c b= .

Example 2

Solve each equation:

a) 2 18y = − b) 83

x= −

c) 2

123a =

Very rarely will we use only one property at a time though!

Example 3

a) 8( 2) 4( 3)y y+ = − b) 8 5 6 3 10z z z− + + = − +

c) 1 1 5 1

6 3 6 2x x− − = +


And again there will be times that you need to translate from word to algebraic

expressions!

Consecutive Integers:

e.g.: 1, 2, 3, 4 ….Expressed by: x, ____________, ____________, …

Consecutive Even Integers:

e.g.: 2, 4, 6 ….Expressed by: x, ____________, ____________, …

Consecutive Odd Integers:

e.g.: 3, 5, 7 ….STILL: x, ____________, ____________, …

Example 4

a) If x is the first two consecutive integers, express the sum of the two integers in

terms of x. Simplify if possible.

b) If x is the first two consecutive odd integers, express the sum of the two integers

in terms of x. Simplify if possible.

c) Two numbers have a sum of 2. If one number is z, express the other number in

terms of z.

d) On a recent car trip, Raymond drove x miles on day one. On day two, he drove

170 miles more than he did on day one. How many miles, in terms of x, did

Raymond drive for both days combined?


2.3 Solving Linear Equations

So in the last section we solve linear equations, and it is still mostly the same.

Example 1

Solve the following linear equations:

a) 6 (5 1) 4a a− − = b) 3(2 3) 5( 4)a a− = +

But now we want to make our lives better by getting rid of the fractions!

Steps for Killing Fractions

1. Find the LCD of ________ fractions.

2. _____________ both sides of the equation, ____________ parentheses, by the

LCD you found in step 1.

3. _____________ to “kill” the fraction!

4. Solve the resulting equation.


Example 2

Solve the following linear equations:

a) 1 3

54 8x x− = b)

1 5( 9) ( 6)

3 6x x− = +

c) 6 5 5

14 4

x x− ++ = −

All of the equations we have solved so far have had one single answer. These types of

equations are called ____________________.

There are also equations that will never be true, they have no solution, and they are

called __________________.

And sometimes equations will be exactly the same on both sides (they are always true),

the solution is _______________________, and these types of equations are called

_____________.

Type Solution

Conditional

All Real Numbers


The nice thing about math is that it is very easy to tell when an equation is true and

when it is not true. Some examples are:

True Statements Not True Statements 3=3 3=5

Example 3

Solve each equation. Indicate its type and its solution:

a) 6( 7) 6 42z z+ = + b) 2

3 13 6

x x− = +


2.4 Introduction to Problem Solving

More Words That Mean

Addition ( + )

Subtraction ( - )

Multiplication ( ∙ )

Division ( ÷ )

Equal Sign (=)

Sum Difference of Product Quotient Equals

Plus Minus Times Divide Gives

Added to subtracted from Multiply Into Is/was/ should be

More than Less than Twice Ratio Yields

Increased by Decreased by Of Divided by Amounts to

Total Represents

Is the same as


Example 1

a) Eight is added to a number and the sum is doubled, the result is 11 less than the

number. Find the number.

b) Three times the difference of a number and 2 is equal to 8 subtracted from twice

a number. Find the integers.

c) A college graduating class is made up of 450 students. There are 206 more girls

than boys. How many boys are in the class?


d) A triangle has three angles, A, B, and C. Angle C is 18° greater than angle B.

Angle A is 4 times angle B. What is the measure of each angle?

(Hint: The sum of the angles of a triangle is 180°).

e) The room numbers of two adjacent hotel rooms are two consecutive odd

numbers. If their sum is 1380, find the hotel room numbers.


2.5 Formulas and Problem Solving

A _________________ is an equation that states a known relationship among multiple quantities (has more than one variable in it).

Formulas you may need

Perimeter: ALWAYS the sum of all of the sides!!

Circumference (circle) = 2 rπ or dπ

Area:

Square = 2s

Rectangle = l w•

Triangle = 1

2bh or

2

bh

Parallelogram = b h•

Trapezoid = 1 2

1( )

2h b b+

Circle = 2rπ

Volume:

Box = l w h• •

Cylinder = 2r hπ

Misc:

Distance: D rt=

Temperature: 5

( 32)9

C F= −

Interest: I prt=


Sometimes you may need to solve a formula for a different variable than what you start

with.

Example 1

Solve for the indicated variable:

a) 1

2A bh= for b b) 4 2S lw wh= + for l

And other times you’ll have to solve an entire word problem using a formula.

Example 2

a) You have decided to fence an area of your backyard for your dog. The length of

the area is 1 meter less than twice the width. If the perimeter of the area is 70

meters, find the length and width of the rectangular area.

b) For the holidays, Chris and Alicia drove 476 miles. They left their house at 7

a.m. and arrived at their destination at 4 p.m. They stopped for 1 hour to rest

and re-fuel. What was their average rate of speed?


2.6 Percents and Problem Solving

The most commonly known way to solve percent problems is:

=

where “is” is the resulting amount, and “of” is the original base number of the problem.

Example 1

a) 544 is what percent of 640?

b) What is 42% of 740?

c) 177 is 20% of what number?

But of course we don’t usually see them this way. Most of the time they will be word

problems where it is harder to decide the “is” and the “of”. Just remember “is” is the

____________________, and “of” is the ______________________ of the problem.

Sales Tax:

Sales tax is some % of the original price.

Price including tax is 100+some % of the original price.

Example 2

Patrick paid $91.16 for golf shoes. If the sales tax rate is 6%, what was the price of the

shoes?


Discounts:

Discount is some % of the original price.

Price after discount is 100-some% of the original price.

Example 3

A jacket is on sale for $53.60. If this represents a 20% discount, what was the original

price of the jacket?

Percent increase or decrease

Percent change = new amount - original amount

100original amount

×

If your answer is positive then it is a percent ________________, and if your answer is

negative then it is a percent __________________.

Example 4

a) The number of minutes on a cell phone bill went from 1200 minutes in March

to1600 minutes in April. Find the percent increase. Round to the nearest whole

percent.


b) In 2004, a college campus had 8,900 students enrolled. In 2005, the same

college campus had 7,600 students enrolled. Find the percent decrease. Round

to the nearest whole percent.


2.7 Further Problem Solving

You will also encounter many problems that involve distance, money, and interest.

Distance Problems

Any time you have a problem that involves distance or travel you will want to use the

distance formula: ___________________.

Example 1

How long will it take a car traveling 60 miles per hour to overtake an activity bus

traveling 45 miles per hour if the activity bus left 2 hours before the car?

D r t

Car

Activity Bus

Money Problems

Whenever you have problems dealing with values then you need to remember that you

have a number of things and a value for those things. For example if I have three

quarters I would say that the value of a quarter is ___________, and the number of

quarters I have is _______, so the total value of my money is __________________.


Example 2

A collection of dimes and quarters and nickels are emptied from a drink machine. There

were four times as many dimes as quarters, and there were ten less nickels than there

were quarters. If the value of the coins was $19.50, find the number of quarters, the

number of dimes, and the number of nickels.

Value each Number Total Value Solution

Quarters

Dimes

Nickels

Total

Example 3

The local church had an ice cream social, and sold tickets for $3 and $2. When the

social was over, 81 tickets had been sold totaling $215. How many of each type of ticket

did the church sell?

Value each Number Total Value Solution

$2 tickets

$3 tickets

Total


Interest Problems

If a problem talks about saving or investing then you have an interest problem and will

need to use the formula: __________________________.

Example 4

Jeff received a year-end bonus of $80,000. He invested some of this money at 8% and

the rest at 10%. If his yearly earned income was $7,300, how much did Jeff invest at

10%?

Principle Rate Time Interest

8% Fund

10% Fund

Total


2.8 Solving Linear Inequalities

Everything we have done so far have been expressions or equations. Now we will solve

inequalities. An inequality is a statement that contains of the symbols: < , >, ≤ or ≥.

Equation Inequality

x = 3

12 7 3y≤ −

Graphing and Interval Notation

When you graph an inequality on a number line or write the solution in interval notation

you need to know some symbols.

Symbol Meaning

( or )

[ or ]

∞

−∞

Example 1

Graph each inequality on a number line and write it in interval notation:

a) 5x ≥ b) 7y <

c) 3

2m− ≥ d)

2

5x >


Set Builder Notation

Another way to write the solution to an inequality is to use set builder notation.

{ } |

Example 2

Write the solution to each inequality in Set Builder Notation:

a) 5x ≥ b) 7y <

c) 3

2m− ≥ d)

2

5x >

Solving Linear Inequalities in One Variable

The nice thing is that all of the properties of equality are the same!! The only thing you

need to be careful of is ____________________ or _____________________ by a

___________________________ number!


Example 3

Use the properties of equality to solve each of the following inequalities. Graph and

write the solution in both set builder and interval notation.

a) 7 12x + ≤ b) 10 3y − > −

c) 83

x≤ − d) 3 24y− >

e) 8 1

( 2) ( 3)21 7x x+ > + f) 18( 2) 21 24a a− − ≥ − +


Compound Inequalities

A compound inequality contains two inequality symbols.

This means 0 4(5 )x≤ − AND 4(5 ) 8x− < .

To solve the compound inequality, perform operations simultaneously to all three parts

of the inequality (left, middle and right).

Example 4

Solve each compound inequality. Graph and write the solution in both set builder and

interval notation.

a) 5 0t− < ≤ b) 12 2 8x− ≤ − <

c) 3 4 9 7x≤ − ≤


Word Problems

Since we solve inequalities almost exactly the same as we solve equations, we will

solve word problems with inequalities almost exactly the same as we solve word

problems with equations. Just be careful with your symbols!

Example 5

a) Chris paid more than $200 for a suit.

b) A candidate for President must be at least 35.

c) The capacity of an elevator is at most 1,500 lbs.

d) The company must have no fewer than 10 programmers.

e) The difference of four times a number and eight is less than two times the

number.

f) One side of a triangle is six times as long as another side and the third side is 8

inches long. If the perimeter can be no more than 106 inches, find the maximum

lengths of the other two sides.

END OF EXAM 1 MATERIAL

2.1 simplifying algebraic expressions - blinn college 0310 notes-exam_1… · 2.1 simplifying...

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