This document defines absolute value and discusses solving absolute value equations in 3 sentences or less: Absolute value is the distance between zero and a number on the number line, and absolute value equations can be solved by rewriting the equation as two separate equations setting the absolute value expression equal to both its positive and negative values. The document provides examples of solving absolute value equations, identifying when an equation has no solution, and applying absolute value to word problems involving acceptable ranges.

1. 4.5 Absolute Value

2. Vocabulary:
• absolute: the distance between zero and a
number on the number line.
• absolute value equation: an equation that
contains an absolute value expression.

3. Example 1 Solve an absolute value equation
Solve x = 7.
SOLUTION
The distance between x and 0 is 7. So, x = 7 or x = –7.
ANSWER The solutions are 7 and –7.

4. Guided Practice for Example 1
Solve the equation.
1. x = 3 ANSWER 3, –3
2. x = 15 ANSWER 15, –15
3. x = 7.8 ANSWER 7.8, – 7.8
3 3 3
4. x = ANSWER ,–
5 5 5

5. Example 2 Multiple Choice Practice
What is the solution for this equation?
2x – 3 = 9
x = – 6 or x = 3 x = – 6 or x = 4
x = – 3 or x = 6 x = – 3 or x = 4
SOLUTION
Rewrite the absolute value equation as two equations.
2x – 3 = 9 Write original equation.
2x – 3 = 9 or 2x – 3 = – 9 Rewrite as two equations.

6. Example 3 Rewrite an absolute value equation
Solve 3 2x + 7 – 5 = 4 .
SOLUTION
First, rewrite the equation in the form ax + b = c .
3 2x + 7 – 5 = 4 Write original equation.
3 2x + 7 = 9 Add 5 to each side.
2x + 7 = 3 Divide each side by 3.
Next, solve the absolute value equation.
2x + 7 = 3 Write absolute value equation.
2x + 7 = 3 or 2x + 7 = – 3 Rewrite as two equations.

7. Guided Practice for Examples 2 and 3
Solve the equation.
5. r – 7 = 9 ANSWER 16, – 2
6. 2 s + 4.1 = 18.9 ANSWER 7.4, – 7.4
7. 4 t + 9 – 5 = 19 ANSWER – 3, – 15

8. Example 4 Decide if an equation has no solutions
Solve 3x + 5 + 6 = – 2 , if possible.
3x + 5 + 6 = – 2 Write original equation.
3x + 5 = – 8 Subtract 6 from each side.
ANSWER
The absolute value of a number is never negative. So,
there are no solutions.

9. Guided Practice for Examples 4 and 5
Solve the equation, if possible.
8. 2 m – 5 + 4 = 2 ANSWER no solution
9. – 3 n + 2 – 7 = – 10 ANSWER – 1, – 3
10. FOOTBALL
An NCAA football must be inflated to an air
pressure of 13 psi with an absolute error of 0.5 psi.
Find the minimum and maximum acceptable air
pressures for an NCAA football.
ANSWER 12.5 psi, 13.5 psi