2.5 Absolute Value Equations and Inequalities Evaluate and graph the absolute value function Solve absolute value equations Solve absolute value inequalities

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.

The Absolute Value Function (2 of 2)

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:

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) = − 2x + 4 (For continuity of the solution, it appears completely on the next slide.)

Example: Analyzing the graph of y = |ax + b| (2 of 2) The graph of y = | −2x + 4| is a reflection of f across the x-axis when y = −2x + 4 is below the x-axis.

Absolute Value Equations (1 of 6) Solutions to |x| = k with k > 0 are given by x = ±k. These concepts can be illustrated graphically.

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

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

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

Absolute Value Equations (5 of 6)

Absolute Value Equations (6 of 6) Let k be a positive number. Then |ax + b| = k is equivalent to ax + b = ±k.

Example: Solving an absolute value equation (1 of 3) Solve the equation |2x + 5| = 2 graphically, numerically, and symbolically. Solution Graph Y1 = abs(2x + 5) and Y2 = 2 Solutions: − 3.5, 1.5

Example: Solving an absolute value equation (2 of 3) Table Y1 = abs(2x + 5) and Y2 = 2 Solutions to y1 = y2 are −3.5 and −1.5.

Example: Solving an absolute value equation (3 of 3)

Absolute Value Inequalities (1 of 2)

Absolute Value Inequalities (2 of 2) Let solutions to |ax + b| = k be s1 and s2, where s1 < s2 and k > 0. 1. |ax + b| < k is equivalent to s1 < x < s2. 2. |ax + b| > k is equivalent to x < s1 or x > s2. Similar statements can be made for inequalities involving ≤ or ≥.

Example: Solving inequalities involving absolute values symbolically Solve the inequality |2x − 5| ≤ 6 symbolically. Write the solution set in interval notation.

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 ≥.

Example: Solving absolute value inequalities Solve the inequality |4 − 5x | ≤ 3. Write your answer in interval notation. Solution |4 − 5x| ≤ 3 is equivalent to the three-part inequality