Inequalities and Absolute Value 8 Chapter Inequalities and Absolute Value

Absolute Value Equations and Inequalities 8.3 Absolute Value Equations and Inequalities 1. Use the distance definition of absolute value. 2. Solve equations of the form |ax + b| = k, for k > 0. 3. Solve inequalities of the form |ax + b| < k and of the form |ax + b| > k, for k > 0. 4. Solve absolute value equations that involve rewriting. 5. Solve equations of the form |ax + b| = |cx + d|. 6. Solve special cases of absolute value equations and inequalities. 7. Solve an application involving relative error.

Use the distance definition of absolute value. Objective 1 Use the distance definition of absolute value.

Absolute Value In Section 1.3, we saw that the absolute value of a number x, written 𝒙 , represents the undirected distance from x to 0 on a number line. For example, the solutions of |x| = 4 are 4 and 4, 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. 4 units from 0 4 units from 0

Solving Absolute Value Equations and Inequalities

Solving Absolute Value Equations and Inequalities

Solve equations of the form |ax + b| = k, for k > 0. Objective 2 Solve equations of the form |ax + b| = k, for k > 0.

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

Solving an Absolute Value Equation (Case 1) Classroom Example 1 Solving an Absolute Value Equation (Case 1) Solve |3x – 4| = 11. Solve the compound equation. 3x – 4 = 11 or 3x – 4 = –11 Check by substituting each solution into the original equation. The solution set is

Objective 3 Solve inequalities of the form |ax + b| < k and of the form |ax + b| > k, for k > 0.

Solving an Absolute Value Equation (Case 2) Classroom Example 2 Solving an Absolute Value Equation (Case 2) Solve |3x – 4| ≥ 11. Solve the inequalities. 3x – 4 ≥ 11 or 3x – 4 ≤ –11 Choose a test point in each interval. The solution set is .

Solving an Absolute Value Equation (Case 3) Classroom Example 3 Solving an Absolute Value Equation (Case 3) Solve |3x – 4| ≤ 11. Solve the three-part inequality. The solution set is

Solving an Absolute Value Equation (Case 2 for ≥) Classroom Example 4 Solving an Absolute Value Equation (Case 2 for ≥) Solve |20 – 2x| ≥ 20. Solve the inequalities. The solution set is

Solve absolute value equations that involve rewriting. Objective 4 Solve absolute value equations that involve rewriting.

Solving an Absolute Value Equation That Requires Rewriting Classroom Example 5 Solving an Absolute Value Equation That Requires Rewriting Solve |3x + 2| + 4 = 15. The check confirms that the solution set is

Solving Absolute Value Inequalities That Require Rewriting Classroom Example 6 Solving Absolute Value Inequalities That Require Rewriting Solve each inequality. a. |x + 2| – 3 > 2 |x + 2| > 5 x + 2 > 5 or x + 2 < –5 x > 3 or x < –7 Solution set:

Solving Absolute Value Inequalities That Require Rewriting (cont.) Classroom Example 6 Solving Absolute Value Inequalities That Require Rewriting (cont.) Solve each inequality. b. |x + 2| – 3 < 2 –5 < x + 2 < 5 –7 < x < 3 Solution set:

Solve equations of the form |ax + b| = |cx + d|. Objective 5 Solve equations of the form |ax + b| = |cx + d|.

Solving |ax + b| = |cx + d| To solve an absolute value equation of the form |ax + b| = |cx + d|, solve the following compound equation. ax + b = cx + d or ax + b = –(cx + d)

Solving an Equation Involving Two Absolute Values Classroom Example 7 Solving an Equation Involving Two Absolute Values Solve |4x – 1| = |3x + 5|.

Solve special cases of absolute value equations and inequalities. Objective 6 Solve special cases of absolute value equations and inequalities.

Special Cases of Absolute Value Case 1 The absolute value of an expression can never be negative—that is, |a| ≥ 0 for all real numbers a. Case 2 The absolute value of an expression equals 0 only when the expression is equal to 0.

Solving Special Cases of Absolute Value Equations Classroom Example 8 Solving Special Cases of Absolute Value Equations Solve each equation. a. |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 b. The expression will equal 0 only if The solution set is {12}.

Solving Special Cases of Absolute Value Inequalities Classroom Example 9 Solving Special Cases of Absolute Value Inequalities Solve each inequality. a. |x| > –1 The absolute value of a number is always greater than or equal to 0. The solution set is (, ). b. |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 .

Solving Special Cases of Absolute Value Inequalities (cont.) Classroom Example 9 Solving Special Cases of Absolute Value Inequalities (cont.) Solve each inequality. c. |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}.

Solve an application involving relative error. Objective 7 Solve an application involving relative error.

Relative error Absolute value is used to find the relative error, or tolerance, in a measurement. If xt represents the expected measurement and x represents the actual measurement, then the relative error in x equals the absolute value of the difference of xt and x, divided by xt.

Solving an Application Involving Relative Error Classroom Example 10 Solving an Application Involving Relative Error Suppose a machine filling quart orange juice cartons is set for a relative error that is no greater than 0.035 oz. How many ounces may a filled carton contain? The filled carton may contain between 30.88 and 33.12 oz, inclusive.