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Section 2.2 Solve Absolute Value Equations and Inequalities |x| |x| = 3 |x| > 3 |x| < 3 Solving Absolute Value Equations and Inequalities Let k be a positive number and p and q be two numbers. To solve |ax + b| = k, solve the compound equation ax + b = k OR ax + b = -k The solution set is usually of the form {p, q}. Solving Absolute Value Equations and Inequalities Let k be a positive number and p and q be two numbers. To solve |ax + b| > k, solve the compound inequality ax + b > k OR ax + b < -k The solution set is usually of the form (-, p) (q, ) Solving Absolute Value Equations and Inequalities Let k be a positive number and p and q be two numbers. To solve |ax + b| < k, solve the compound inequality -k < ax + b < k The solution set is usually of the form (p, q) |x| |x| = 3 |x| > 3 |x| < 3 Solve each equation and graph the solution set. |x + 2| = 3 x + 2 = -3 x + 2 - 2 = -3 - 2 x=-5 x+2=3 x+2-2=3-2 x=1 x ={-5, 1} Solve each equation and graph the solution set. |x + 2| > 3 x + 2 < -3 x + 2 - 2 < -3 - 2 x<-5 x+2>3 x+2-2>3-2 x>1 (-, -5) (1, ) ) ( Solve each equation and graph the solution set. |x + 2| < 3 -3 < x + 2 < 3 -3 - 2 < x + 2 - 2 < 3 - 2 -5 < x < 1 (-5, 1) ( ) Cautions • These methods apply when the constant is alone and positive. • Absolute equations and |ax+b| > k translate into OR compound statements. • |ax+b| < k translate into AND compound statements. • An OR statement cannot be written into three parts. |ax + b| = |cx + d| To solve an equation of this form solve the compound equation ax + b = cx + d OR ax + b = -(cx + d) |4r - 1| = |3r + 5| 4r 1 3r 5 4r 1 1 3r 5 1 4r 3r 6 4r 1 (3r 5) 4r 1 3r 5 4r 1 1 3r 5 1 4r 3r 3r 3r 6 4r 3r 4 4r 3r 3r 3r 4 r 6 7r 4 4 r 6, 4 r 7 7 |6x + 7| = - 5 Since the absolute value of an expression can never be negative, there are no solutions to this equation. Solution set = Special Cases |x| > -1 (-, ) |y| < -5 Solution set = |k + 2| < 0 k = -2 Matching E |x| = 5 |x| < 5 C |x| > 5 D |x| < 5 B |x| > 5 A A ] [ [ ] B ( ) C ) ( D E Solve |2y + 3|= 19 2y + 3 = 19 2y = 16 y=8 2y + 3 = -19 2y = -22 y = -11 y = {8, -11} Solve |3x - 1|> 8 3x - 1 > 8 3x > 9 x>3 ] [ 3x - 1 < -8 3x < -7 7 x 3 7 x (, ] [3, ) 3 Section 2.2 Homework Page 30 #1-25 Odd Page 31 #17 – 23 Odd Review for Quiz Page 44 #1 – 15 Odd Page 30 #1-25 Odd