6.5

Warm UpWarm Up

Lesson QuizLesson Quiz

Lesson PresentationLesson Presentation

Solve Absolute Value Equations

6.5 Warm-Up

ANSWER 12, 12

ANSWER 1

2. Evaluate |x| – 2 when x = –3.

1. For a = –12, find, –a and |a|.

The change in elevation as a diver explored a reef was –0.5 foot, 1.5 feet, –2.5 feet, and 2.25 feet. Which change in elevation had the greatest absolute value?

ANSWER –2.5 ft

3.

6.5 Example 1

SOLUTION

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

ANSWER

The solutions are 7 and –7.

Solve x = 7.

6.5 Guided Practice

Solve (a) |x| = 3 and (b) |x| = 15.

ANSWER 3, –3a.

15, –15b.

6.5 Example 2

SOLUTION

Rewrite the absolute value equation as two equations. Then solve each equation separately.

x – 3 = 8 Write original equation.

x – 3 = 8 or x – 3 = –8 Rewrite as two equations.

x = 11 or x = –5 Add 3 to each side.

ANSWER

The solutions are 11 and –5. Check your solutions.

x – 3 = 8.Solve

6.5 Example 2

|x – 3| = 8 |x – 3| = 8

CHECK

Substitute for x.

Subtract.

Simplify. The solution checks.

|11 – 3| = 8 |–5 – 3| = 8? ?

| 8| = 8 |–8| = 8??

Write original inequality.

8 = 8 8 = 8

6.5 Example 3

SOLUTION

First, rewrite the equation in the form ax + b = c.

3 2x – 7 – 5 = 4

3 2x – 7 = 9

2x – 7 = 3

Write original equation.

Add 5 to each side.

Divide each side by 3.

3 2x – 7 – 5 = 4.Solve

6.5 Example 3

Next, solve the absolute value equation.

2x – 7 = 3

2x – 7 = 3 or 2x – 7 = –3

2x = 10 or 2x = 4

x = 5 or x = 2

Write absolute value equation.

Rewrite as two equations.

Add 7 to each side.

Divide each side by 2.

ANSWER

The solutions are 5 and 2.

6.5 Guided Practice

Solve the equation.r – 7 = 92.

16, –2ANSWER

2 s + 4.1 = 18.93.

7.4, –7.4ANSWER

4 t + 9 – 5 = 194.

–3, –15ANSWER

6.5 Example 4

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.

3x + 5 + 6 = –2, if possible.Solve

6.5 Guided Practice

ANSWER no solution

5. 2 m – 5 + 4 = 2

Solve the equation, if possible

6. –3 n +2 –7 = –10

ANSWER 1, 3

6.5

Absolute Deviation

The absolute deviation of a number x from a given value is the absolute value of the difference of x and the given value:

Absolute Deviation =

6.5 Example 5

BASKETBALLS

Before the start of a professional basketball game, a basketball must be inflated to an air pressure of 8 pounds per square inch (psi) with an absolute error of 0.5 psi.

Absolute error is the absolute deviation of a measured value from an accepted value.

Find the minimum and maximum acceptable air pressures for the basketball.

6.5 Example 5

SOLUTION

Let p be the air pressure (in psi) of a basketball. Write a verbal model. Then write and solve an absolute value equation.

0.5 = p – 8

6.5 Example 5

p – 80.5 =

p 80.5 = – or p 8–0.5 = –

p8.5 = or p7.5 =

Write original equation.

Rewrite as two equations.

Add 8 to each side.

ANSWER

The minimum and maximum acceptable pressures are 7.5 psi and 8.5 psi.

6.5 Guided Practice

7. A volleyball league is preparing a two minute radio ad to announce tryouts. The ad has an absolute deviation of 0.05 minute. Find the minimum and the maximum acceptable times the radio ad can run.

Minimum: 1.95 minMaximum: 2.05 min

ANSWER

6.5 Lesson Quiz

Solve the equation, if possible.

ANSWER –9, 17

ANSWER no solutions

1. 3| x – 4 | = 39

2. | x + 2 | + 7 = 3

ANSWER –5, 11

ANSWER no solutions

3. | 2x – 6 | – 18 = – 2

4. –2 | x – 5 | + 7 = 12

6.5 Lesson Quiz

A pattern for a 26-inch skirt has an absolute deviation of 1.5 inches. Find the minimum and maximum skirt lengths that can be made from the pattern.

5.

ANSWER minimum : 24.5 in.; maximum 27.5 in.