1. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 1 Chapter 2 Linear Functions and Equations

2. 2 Copyright © 2014, 2010, 2006 Pearson Education, Inc. Absolute Value Equations and Inequalities ♦ Evaluate and graph the absolute value function ♦ Solve absolute value equations ♦ Solve absolute value inequalities 2.5

3. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 3 Absolute Value Function The graph of y = |x|. V-shaped Cannot be represented by single linear function

4. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 4 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:

5. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 5 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) = –2x + 4 (For continuity of the solution, it appears completely on the next slide.) Example: Analyzing the graph of y = |ax + b|

6. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 6 The graph of y = |–2x + 4| is a reflection of f across the x-axis when y = –2x + 4 is below the x-axis. Example: Analyzing the graph of y = |ax + b|

7. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 7 Absolute Value Equations Solutions to |x| = k with k > 0 are given by x = ±k. Solutions to |ax + b| = k are given by ax + b = ±k. These concepts can be illustrated visually.

8. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 8 Absolute Value Equations Two solutions |ax + b| = k, for k > 0

9. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 9 Absolute Value Equations One solution |ax + b| = k, for k = 0

10. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 10 Absolute Value Equations No solution |ax + b| = k, for k < 0

11. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 11 Absolute Value Equations Let k be a positive number. Then |ax + b| = k is equivalent to ax + b = ±k.

12. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 12 Solve the equation |2x + 5| = 2 graphically, numerically, and symbolically. Graph Y 1 = abs(2X + 5) and Y 2 = 2 Solution Solutions: –3.5, –1.5 Example: Solving an equation with technology

13. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 13 Solutions to y 1 = y 2 are –3.5 and –1.5. Table Y 1 = abs(2x + 5) and Y 2 = 2 Example: Solving an equation with technology

14. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 14 Symbolic: Example: Solving an equation with technology

15. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 15 Absolute Value Inequalities Solutions |ax + b| = k labeled s 1 and s 2 and the graph of y = |ax + b| is below y = k between s 1 and s 2 or when s 1 < x < s 2. Solution to |ax + b| < k is in green.

16. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 16 Absolute Value Inequalities Solutions |ax + b| = k labeled s 1 and s 2 and the graph of y = |ax + b| is above y = k to left of s 1 and right of s 2 or x s 2. Solution to |ax + b| > k is in green.

17. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 17 Absolute Value Inequalities Let solutions to |ax + b| = k be s 1 and s 2, where s 1 0. 1.|ax + b| < k is equivalent to s 1 < x < s 2. 2.|ax + b| > k is equivalent to x s 2. Similar statements can be made for inequalities involving ≤ or ≥.

18. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 18 Solve the inequality |2x – 5| ≤ 6. Write the solution set in interval notation. Solve |2x – 5| = 6 or 2x – 5 = ±6 Solution Solution set: Example: Solving inequalities involving absolute values symbolically

19. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 19 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. Similar statements can be made for inequalities involving ≤ or ≥.

20. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 20 Solve the inequality |4 – 5x | ≤ 3. Write your answer in interval notation. |4 – 5x| ≤ 3 is equivalent to the three-part inequality Solution In interval notation, solution is. Example: Using an alternative method