2 5 Absolute Value Equations and Inequalities Evaluate

2. 5 Absolute Value Equations and Inequalities • Evaluate and graph the absolute value function • Solve absolute value equations • Solve absolute value inequalities Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

The Absolute Value Function (1 of 2) The absolute value function is defined by ƒ(x) = |x|. The following See the Concept (next slide) describes many of the properties of this function. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

The Absolute Value Function (2 of 2) Copyright © 2018, 2014, 2010 Pearson Education,

Absolute Value Function Alternate Formula That is, regardless of whether a real number x is positive or negative, the expression equals the absolute value of x. Examples: Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Example: Analyzing the graph of y = |ax + b| (1 of 2) For the linear function f, graph y = f (x) and y = |f (x)| separately. Discuss how the absolute value affects the graph of f. f(x) = − 2 x + 4 (For continuity of the solution, it appears completely on the next slide. ) Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Example: Analyzing the graph of y = |ax + b| (2 of 2) The graph of y = | − 2 x + 4| is a reflection of f across the x-axis when y = − 2 x + 4 is below the x-axis. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Absolute Value Equations (1 of 6) Solutions to |x| = k with k > 0 are given by x = ±k. These concepts can be illustrated graphically. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Absolute Value Equations (2 of 6) Solving |x| = 5 Graphically Two solutions: −

Absolute Value Equations (3 of 6) Solving |x| = 5 Graphically Two solutions: −k,

Absolute Value Equations (4 of 6) Solutions to |ax + b| = k are

Absolute Value Equations (5 of 6) Copyright © 2018, 2014, 2010 Pearson Education, Inc.

Absolute Value Equations (6 of 6) Let k be a positive number. Then |ax

Example: Solving an absolute value equation (1 of 3) Solve the equation |2 x + 5| = 2 graphically, numerically, and symbolically. Solution Graph Y 1 = abs(2 x + 5) and Y 2 = 2 Solutions: − 3. 5, 1. 5 Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Example: Solving an absolute value equation (2 of 3) Table Y 1 = abs(2 x + 5) and Y 2 = 2 Solutions to y 1 = y 2 are − 3. 5 and − 1. 5. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Example: Solving an absolute value equation (3 of 3) Copyright © 2018, 2014, 2010

Absolute Value Inequalities (1 of 2) Copyright © 2018, 2014, 2010 Pearson Education, Inc.

Absolute Value Inequalities (2 of 2) Let solutions to |ax + b| = k be s 1 and s 2, where s 1 < s 2 and k > 0. 1. |ax + b| < k is equivalent to s 1 < x < s 2. 2. |ax + b| > k is equivalent to x < s 1 or x > s 2. Similar statements can be made for inequalities involving ≤ or ≥. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Example: Solving inequalities involving absolute values symbolically Solve the inequality |2 x − 5| ≤ 6 symbolically. Write the solution set in interval notation. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Absolute Value Inequalities (Alternative Method) Let k be a positive number. 1. |ax + b| < k is equivalent to −k < ax + b < k. 2. |ax + b| > k is equivalent to ax + b < −k or ax + b > −k. Similar statements can be made for inequalities involving ≤ or ≥. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Example: Solving absolute value inequalities Solve the inequality |4 − 5 x | ≤ 3. Write your answer in interval notation. Solution |4 − 5 x| ≤ 3 is equivalent to the three-part inequality Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved