9.2 Absolute Value Equations and Inequalities

9.2 Absolute Value Equations and InequalitiesUse the distance definition of absolute value.Objective 1 Slide 9.2- 2The 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 9.2- 3Use the distance definition of absolute value.

Slide 9.2- 4Use the distance definition of absolute value.Solve equations of the form |ax + b| = k, for k > 0.

Objective 2 Slide 9.2- 5Remember that because absolute value refers to distance from the origin, an absolute value equation will have two parts. Slide 9.2- 6Use the distance definition of absolute value.Solve |3x 4| = 11.

3x 4 = 11 or 3x 4 = 11 3x 4 + 4 = 11 + 4 3x 4 + 4 = 11 + 4

3x = 73x = 15

x = 5

Check by substituting and 5 into the original absolute value equation to verify that the solution set isSlide 9.2- 7CLASSROOM EXAMPLE 1Solving an Absolute Value EquationSolution:

Solve inequalities of the form |ax + b| < k and of the form |ax + b| > k, for k > 0.Objective 3 Slide 9.2- 8Solve |3x 4| 11.

3x 4 11 or 3x 4 11 3x 4 + 4 11 + 4 3x 4 + 4 11 + 4 3x 7 3x 15 x 5

Check the solution. The solution set is The graph consists of two intervals. [8-4-202468-5-137-351-5]Slide 9.2- 9CLASSROOM EXAMPLE 2Solving an Absolute Value Inequality with >Solution:

Solve |3x 4| 11.

11 3x 4 11 11 + 4 3x 4 11+ 4 7 3x 15

Check the solution. The solution set is

The graph consists of a single interval.8-4-202468-5-137-351-5Slide 9.2- 10CLASSROOM EXAMPLE 3Solving an Absolute Value Inequality with 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 9.2- 11Solve inequalities of the form |ax + b| < k and of the form |ax + b| > k, for k > 0.

Solve absolute value equations that involve rewriting.

Objective 4 Slide 9.2- 12Solve |3x + 2| + 4 = 15.

First get the absolute value alone on one side of the equals sign.|3x + 2| + 4 = 15 |3x + 2| + 4 4 = 15 4 |3x + 2| = 11

3x + 2 = 11 or 3x + 2 = 11 3x = 133x = 9 x = 3

The solution set isSlide 9.2- 13CLASSROOM EXAMPLE 4Solving an Absolute Value Equation That Requires RewritingSolution:

Solve the inequality.

|x + 2| 3 > 2

|x + 2| 3 > 2

|x + 2| > 5 x + 2 > 5 or x + 2 < 5 x > 3 x < 7

Solution set: (, 7) (3, )Slide 9.2- 14CLASSROOM EXAMPLE 5Solving Absolute Value Inequalities That Require RewritingSolution:Solve the inequality.

|x + 2| 3 < 2

|x + 2| < 5 5 < x + 2 < 5

7 < x < 3Solution set: (7, 3)Slide 9.2- 15CLASSROOM EXAMPLE 5Solving Absolute Value Inequalities That Require Rewriting (contd)Solution:Solve equations of the form |ax + b| = |cx + d| .Objective 5 Slide 9.2- 16Solving |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 9.2- 17Solve equations of the form |ax + b| = |cx + d|.Solve |4x 1| = |3x + 5|.

4x 1 = 3x + 5 or 4x 1 = (3x + 5)

4x 6 = 3x or 4x 1 = 3x 5 6 = x or 7x = 4

x = 6 or

Check that the solution set is

Slide 9.2- 18CLASSROOM EXAMPLE 6Solving an Equation with Two Absolute ValuesSolution:

Solve special cases of absolute value equations and inequalities.Objective 6 Slide 9.2- 19Special 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 9.2- 20Solve special cases of absolute value equations and inequalities.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 9.2- 21CLASSROOM EXAMPLE 7Solving Special Cases of Absolute Value EquationsSolution:

The expression will equal 0 only if

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

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 Add 2 to each side.

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 9.2- 22CLASSROOM EXAMPLE 8Solving Special Cases of Absolute Value InequalitiesSolution: