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Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectiv es 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute value. Solve equations of the form |ax + b| = k, for k > 0. Solve inequalities of the form |ax + b| < k and of the form |ax + b| > k, for k > 0. Solve absolute value equations that involve rewriting. Solve equations of the form |ax + b| = |cx + d| . Solve special cases of absolute value equations and inequalities. 2. 7

Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute

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Page 1: Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

1

Objectives

2

6

5

3

4

Absolute Value Equations and Inequalities

Use the distance definition of absolute value.

Solve equations of the form |ax + b| = k, for k > 0.

Solve inequalities of the form |ax + b| < k and of the form |ax + b| > k, for k > 0.

Solve absolute value equations that involve rewriting.

Solve equations of the form |ax + b| = |cx + d| .

Solve special cases of absolute value equations and inequalities.

2.7

Page 2: Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Use the distance definition of absolute value.

Objective 1

Slide 2.7- 2

Page 3: Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

The absolute value of a number x, written |x|, is the distance from x to 0 on the number line.

For example, the solutions of |x| = 5 are 5 and 5, as shown below. We need to understand the concept of absolute value in order to solve equations or inequalities involving absolute values. We solve them by solving the appropriate compound equation or inequality.

Distance is 5, so |5| = 5.

Distance is 5, so |5| = 5.

Slide 2.7- 3

Use the distance definition of absolute value.

Page 4: Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 2.7- 4

Use the distance definition of absolute value.

Page 5: Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Solve equations of the form |ax + b| = k, for k > 0.

Objective 2

Slide 2.7- 5

Remember that because absolute value refers to distance from the origin, an absolute value equation will have two parts.

Page 6: Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 2.7- 6

Use the distance definition of absolute value.

Page 7: Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Solve:

Slide 2.7- 7

CLASSROOM EXAMPLE 1

Solving an Absolute Value Equation

|3x – 4| = 11

|-2x + 5| = -7

Page 8: Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Solve inequalities of the form |ax + b| < k and of the form |ax + b| > k, for k > 0.

Objective 3

Slide 2.7- 8

Page 9: Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Solve:

|3x – 4| 11.

Slide 2.7- 9

CLASSROOM EXAMPLE 2

Solving an Absolute Value Inequality with >

Page 10: Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Solve:

|4x - 3| > 9

Slide 2.7- 10

CLASSROOM EXAMPLE 2

Solving an Absolute Value Inequality with >

Page 11: Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Solve:

|3x – 4| ≤ 11

Slide 2.7- 11

CLASSROOM EXAMPLE 3

Solving an Absolute Value Inequality with <

Page 12: Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Solve:

|2x + 3| < 15

Slide 2.7- 12

CLASSROOM EXAMPLE 3

Solving an Absolute Value Inequality with <

Page 13: Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

When solving absolute value equations and inequalities of the types in Examples 1, 2, and 3, remember the following:

1. The methods describe apply when the constant is alone on one side of the equation or inequality and is positive.

2. Absolute value equations and absolute value inequalities of the form |ax + b| > k translate into “or” compound statements.

3. Absolute value inequalities of the form |ax + b| < k translate into “and” compound statements, which may be written as three-part inequalities.

4. An “or” statement cannot be written in three parts.

Slide 2.7- 13

Solve inequalities of the form |ax + b| < k and of the form |ax + b| > k, for k > 0.

Page 14: Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Solve absolute value equations that involve rewriting.

Objective 4

Slide 2.7- 14

Page 15: Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Solve:

|3x + 2| + 4 = 15

Slide 2.7- 15

CLASSROOM EXAMPLE 4

Solving an Absolute Value Equation That Requires Rewriting

Page 16: Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Solve the inequality.

|x + 2| – 3 > 2

Slide 2.7- 16

CLASSROOM EXAMPLE 5

Solving Absolute Value Inequalities That Require Rewriting

Page 17: Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Solve the inequality.

|x + 2| – 3 < 2

Slide 2.7- 17

CLASSROOM EXAMPLE 5

Solving Absolute Value Inequalities That Require Rewriting (cont’d)

Page 18: Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Solve equations of the form |ax + b| = |cx + d| .

Objective 5

Slide 2.7- 18

Page 19: Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Solving |ax + b| = |cx + d|To solve an absolute value equation of the form

|ax + b| = |cx + d| ,

solve the compound equation

ax + b = cx + d or ax + b = (cx + d).

Slide 2.7- 19

Solve equations of the form |ax + b| = |cx + d|.

Page 20: Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Solve:

4x – 1| = |3x + 5|

Slide 2.7- 20

CLASSROOM EXAMPLE 6

Solving an Equation with Two Absolute Values

Page 21: Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Solve special cases of absolute value equations and inequalities.

Objective 6

Slide 2.7- 21

Page 22: Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Special Cases of Absolute Value1. The absolute value of an expression can never be negative; that

is, |a| 0 for all real numbers a.

2. The absolute value of an expression equals 0 only when the expression is equal to 0.

Slide 2.7- 22

Solve special cases of absolute value equations and inequalities.

Page 23: Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Solve each equation.

|6x + 7| = –5

The absolute value of an expression can never be negative, so there are no solutions for this equation. The solution set is .

Slide 2.7- 23

CLASSROOM EXAMPLE 7

Solving Special Cases of Absolute Value Equations

13 0

4x

The expression will equal 0 only if

The solution of the equation is 12.

The solution set is {12}, with just one element.

13

4x 1

34x

12.x

Page 24: Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 6 5 3 4 Absolute Value Equations and Inequalities Use the distance definition of absolute

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Solve each inequality.|x | > – 1

The absolute value of a number is always greater than or equal to 0. The solution set is (, ).

|x – 10| – 2 ≤ –3 |x – 10| ≤ –1 There is no number whose absolute value is less than –1, so the inequality has no solution. The solution set is .

|x + 2| ≤ 0

The value of |x + 2| will never be less than 0. |x + 2| will equal 0 when x = –2. The solution set is {–2}.

Slide 2.7- 24

CLASSROOM EXAMPLE 8

Solving Special Cases of Absolute Value Inequalities