4.5 Absolute Value

Vocabulary:

• absolute: the distance between zero and a number on the number line.

• absolute value equation: an equation that contains an absolute value expression.

Solve an absolute value equation

Solve

Example 1

= 7.x

SOLUTION

The distance between x and 0 is 7. So, or= 7x 7.–=x

The solutions are 7 and –7.ANSWER

Guided Practice for Example 1

1.

Solve the equation.

= 3x ANSWER 3, 3–

2. = 15x ANSWER 15, 15–

3. = 7.8x ANSWER 7.8, 7.8–

4. =x5

3ANSWER ,

5

3–

5

3

Example 2 Multiple Choice Practice

What is the solution for this equation?

SOLUTION

Rewrite the absolute value equation as two equations.

= 3x6 or–=x

= 6x3 or–=x

= 4x6 or–=x

= 4x3 or–=x

=32x – 9

=32x – 9 Write original equation.

=32x – 9 Rewrite as two equations.=32x – 9–or

Solve .72x + 5 = 4–3

SOLUTION

First, rewrite the equation in the form .bax + = c

Write original equation.72x + 5 = 4–3

Add 5 to each side.72x + = 93

Rewrite an absolute value equationExample 3

Divide each side by 3.72x + = 3

Rewrite as two equations.=72x + 3 =72x + 3–or

Write absolute value equation.

Next, solve the absolute value equation.

72x + = 3

ANSWER 7.4, 7.4–

ANSWER 16, 2–

Guided Practice for Examples 2 and 3

5.

Solve the equation.

=7r – 9

s 4.1 = 18.92 +6.

9t + 5 = 19–47. ANSWER 15–3,–

Decide if an equation has no solutions

Solve , if possible.

Example 4

53x + 6 =+ 2–

Write original equation.53x + 6 =+ 2–

Subtract 6 from each side.53x + = 8–

The absolute value of a number is never negative. So, there are no solutions.

ANSWER

Guided Practice for Examples 4 and 5

8.

Solve the equation, if possible.

5m 4 =+ 2–2

9. 2n 7 =3– + – 10–

10. 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.

FOOTBALL

ANSWER no solution

ANSWER 12.5 psi, 13.5 psi

ANSWER 1,– 3–