chapter 1: equations and inequalitiesmrsmasonmath.weebly.com/.../3/5963714/chp_1_online.pdf · chp...

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Chapter 1: Equations and inequalities1.1 apply properties of real numbers

Real Number System

NATURAL

WHOLE

INTEGERS

IRRATIONALSRATIONALS

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Natural Numbers

Counting Numbers1, 2, 3, 4, 5, ...

Example of where you have seen Natural Numbers used.

∞

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Whole Numbers

0 + All of the Natural Number 0, 1, 2, 3, 4, 5, ...

Example of where you have seen Whole Numbers used.

∞

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Integers

All of the Whole Numbers + all of the opposites of the Natural Numbers

. . . , -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, . . .

Example of where you have seen Integers used.

‐∞ ∞

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Rationals

• All Natural, Whole, and Integers• Any number you can write as a fraction

where a & b are integers with b≠0

ab

• Any terminating decimal (0.5, 7.13, -6.876, -24.45)

• Any repeating decimal (‐10., 2., 3.6262…, ‐2.12333…)

• Square roots of Perfect Square Numbers (1, 4, 9, 16, 25, ...)

Example of where you have seen Rational Numbers used.

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Irrationals

• Any number that is NOT Rational• Decimals that do not terminate AND do not repeat• "CRAZY NUMBERS"• Square roots of non-perfect square numbers (√2,

√23, √30, √55)• Can be positive or negative

Example of where you have seen Irrational Numbers used.

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Graph real numbers on a number line

­1 0­2­3­4­5 1 2 3 4 5

Graph the real number on a number line

Which list shows the numbers in increasing order?

a) ­0.5, 1.5, ­2, ­0.75, √7             b.) ­0.5, ­2, ­0.75, 1.5, √7

c) ­2, ­0.75, ­0.5, 1.5, √7             d.) √7, 1.5, ­0.5, ­0.75, ­2

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Properties of addition and multiplication 

let a, b, and c be real numbers

Property

Closure

Commutative

Associative

Identity

Inverse

Addition Multiplication

a + b is a real number.

a + b = b + a

(a + b) + c = a + (b + c)

a + 0 = a, 0 + a = a

a + (­a) = 0

ab is a real number.

ab = ba

(ab)c = a(b c)

1 a = a, a 1 = a

a = 1, a ≠ 0

The following property involves both addition and multiplication

Distributive: a(b+c) = ab + ac

Defining Subtraction and Division

Subtraction is defined as adding the opposite. The opposite, or additive inverse, or any number b is ­b. If b is positive, then ­b is negative. If b is negative, then ­b is positive.

Division is defined as multiplying by the reciprocal. The reciprocal, or multiplicative inverse, of any nonzero number b is 1/b.

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Identify the property that the statement illustrates 

Commutative

Associative

Identity

Inverse

Distributive

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Unit analysis and conversions 

You work 4 hours and earn $36You travel 2.5 hours at 50 mph. How far did you go?

You drive 45 miles per hour.  What is your speed in feet per second?

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1.2 Evaluate and Simplify Algebraic expressions

A numerical expression consists of numbers, operations and grouping symbols.  An expression formed by repeated multiplication of the same factor is a power.

73base

exponentPower

Terms and Coefficients

3x2 + 5x + 7

coefficients

variable termsconstant terms

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Evaluating different expressions

1.

2.

3.

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1.3 Solve Linear equations

An equation is a statement that two expressions are equal.  A linear equation in one variable is an equation that can be written in the form ax+b = 0 where a and b are constants and a≠0.

example 1:

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example 2:

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example 3:

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Car Wash: It takes you 8 minutes to wash a car and it takes a friend 6 minutes to wash a car. How long does it take the two of you to wash 7 cars if you work together?

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1.4 Rewrite formulas and equationsA formula is an equation that relates two or more quantities, usually represented by variables.  Some common formulas are shown below.

Distance

Temperature

Area of a Triangle

Area of a Rectangle

Perimeter of a Rectangle

Area of a trapezoid

Area of a circle

Circumference of a circle

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example 1: The formula for the distance d between opposite vertices of a regular hexagon is below. Where a is the distance between opposite sides. Solve the formula for a.

example 1.5: Write the formula giving the are of a circle in terms of its circumference.

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example 2: Rewrite linear/nonlinear equations

Linear: Solve for y

Nonlinear: Solve for y

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example 3: Solve for y, leave your answer in simplest form

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1.6 Solve Linear Inequalities

Graphing inequalities:

or

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examples: Solving and graphing inequalities

or

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example: A monitor lizard has a temperature that ranges from 18oC to 34oC. Write the range of temperatures as a compound inequality. Then write an inequality giving the temperature range in degrees Fahrenheit.

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1.7 Solve Linear Inequalities

Interpreting Absolute Value Equations

Equation:

Meaning:

Solution:

The distance between x and b is k

or

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Example: Solve absolute value equations

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Example: Check for extraneous solutionsExtraneous Solutions:  It is possible for a solution to be extraneous.  An extraneous solution is an apparent solution that must be rejected because it does not satisfy the original equation. 

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Example: Solve and graph an absolute value inequality

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