Ch 2.2

Objective:

To solve problems involving absolute value of numbers or variables.

Definitions

Opposite (-)

Take the negative of a value to create the opposite of that number.

For example: The opposite of 4 is -4. The opposite of -4 is –(-4) = 4.

Absolute Value (| |)

Represents the distance (which is always positive) from 0 on a number line.

For example: The absolute value of -2 is 2 which is written as |-2| = 2

Solving Absolute Value Problems

If you remove the absolute value sign, you must replace it with BOTH a positive sign AND a negative sign.

 For example: Solve |x| = 2

This results in two equations: +x = 2 and –x = 2

Try These

• What is the opposite of 0? ______

• What is the opposite of 10? _____

• What is the opposite of -6? ______

• What is the absolute value of 0 written as |0|? ________

• What is the absolute value of 7 written as |7|? ________

• What is the absolute value of -5 written as |-5|? _______

• What is the opposite of the absolute value of -2

written as -|-2|? _________

Absolute value:The distance from

zero on the number line.

-5 -4 -3 -2 -1 0 1 2 3 4 5

3 =

− =5

− =10

12 =

− =7

−− =23

0 =

− =5 4.

− =16

3

5

10

12

-7

-23

0

5.4

-16

Opposites vs. Absolute Value

Given Number Opposite Absolute Value

8 - 8 8 8=

-24 +24 − =24 24

-3.5 +3.5 − =35 35. .

412

−412

41

24

1

2=

t = or -34

Solve each equation below.

1)

2)

3)

4)

5)

6)

x =10

x =4

x =0

x =−5

− =−x 14

t =34

x = 10 or -10

x = 4 or - 4

x = 0

“no solution”

x = 14 or - 14

34

Determine whether each statement is true always,sometimes, or never for all real numbers.

1)

2)

3)

4)

5)

6)

x x=

− ≤x x

x x>−

x < 0

− < −x x

x x=−

sometimes

always

sometimes

never

sometimes

always

Velocity vs. Speed

Velocity - Indicates speed and direction.

Speed - The absolute value of velocity.

Example:

A helicopter descends at 50 feet/second.

A) What is its velocity?

B) What is its speed?

-50 ft./sec.

+50 ft./sec.

Counterexamples

To prove a statement true, it must be proven true for all examples - difficult!

Counterexample - An example that provesa statement false.

Statement: All pets are furry.

Counterexample: Goldfish.

Statement:

Counterexample:

€

x > x

€

x = 0

Determine whether each statement is true or falsefor all real numbers. If it is false, find a counter-example that proves it is false.

1)

2)

3)

4)

5)

6)

x > 0

x x=

− =−x x

x x≥−

x x≥−

x x=

False, x = 0

False, x = -7

False, x = -5

True

True

True