Welcome: Solve each equation and inequality: |3x + 5| = -5 No solution; (2) No solution; (3) All real numbers

HW Key: Solving Absolute Value Equations & Inequalities WKS {4, 12} (1, 7) (-, -10] U [-4, ) [-0.5, 4.5] {-10, 4} (-, 0] U [6, ) [-4.25, .25] (-2, 6)

Solving Equations and Inequalities Involving Two Absolute Value Expressions

Objectives and HW: The students will be able to solve equations and inequalities involving two absolute value expressions. HW: p. 125: 67-69 all –> Do (a) – (c) for each

Solving |ax + b| = |cx + d| To solve the equation |ax + b| = |cx + d| analytically, solve the compound equation: ax + b = cx + d or ax + b = -(cx + d)

Example 1: Solve analytically: |2x + 3| = |4x – 7| Answer: {5, 2/3}

Example 2: Solve graphically: |2x + 3| = |4x – 7| Let y1 = |2x + 3| and y2 = |4x – 7|. Since y1 = y2, then y3 = y1 – y2 y3 = |2x + 3| – |4x – 7| Put y3 in the calculator and graph. Then find the zeros (2nd Trace, Zero) of y3, i.e. find the x-intercepts. Note: Absolute value is found under Math, Num, 1:abs(. Answer: {5, 2/3}

Example 3: Solve graphically: |2x + 3| > |4x – 7| Using the same graph in example 2, simply analyze f(x) > 0 where f(x) = |2x + 3| – |4x – 7| Answer: (2/3, 5); Note: Vertex is at about (1.75, 6.5).

Example 4: Solve graphically: |2x + 3| < |4x – 7| Using the same graph in example 2, simply analyze f(x) < 0 where f(x) = |2x + 3| – |4x – 7| Answer: (-, 2/3) U (5, ); Note: Vertex is at about (1.75, 6.5).

Students Try!!! 1) Solve analytically: |x + 6| = |3x – 5|. Solve graphically: a) |x + 6| > |3x – 5| b) |x + 6| < |3x – 5| Answers: (1) {-.25, 5.5}; (2a) (-.25, 5.5); (2b) (-, -.25) U (5.5, )