2 7 Solving absolute Value inequality Objective Solving

2. 7 Solving absolute Value inequality Objective: Solving absolute value inequality Case 1: |x| < a Case 2: |x| > a

Definition of Absolute value Absolute Value of a number is… The distance of that number from zero. |-3| Read as “The distance of -3 from zero” |-3| = 3 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 |+5| Read as “The distance of +5 from zero” |+5| = 5 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Solve and graph |x| = 7 x=7 OR Read as “The distance of x from zero is 7” x = -7 Graph -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Solve and graph x+4=5 -4 -4 x = 1 |x + 4 | = 5 OR Read as “The distance of x +4 from zero is 5” x + 4 = -5 -4 -4 x = -9 Graph -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Solve and graph |2 x - 3| - 11 = - 4 Before start solving, you MUST ISOLATE the absolute value expression on one side |2 x - 3| - 11 = - 4 +11 |2 x - 3| = 7 2 x - 3 = 7 +3 OR 2 x - 3 = -7 +3 2 x = 10 2 x=5 2 +3 +3 2 x = - 4 2 2 x = -2 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Case 1: |x| < a Solve and graph -a < x < a |x| < 6 Read as “The distance of x from zero is less than 6” -6 < x < 6 Graph -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Solve and graph |x + 4| ≤ 2 Read as “The distance of x + 4 from zero is less than or equal to 2” -2 ≤ x + 4 ≤ 2 -4 -4 -4 -6 ≤ x ≤ -2 Graph -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Solve and graph |x - 1| -5 < - 4 Before start solving, you MUST ISOLATE the absolute value expression on one side |x - 1| -5 < - 4 +5 +5 | x - 1| < 1 -1 < x – 1 < 1 +1 +1 +1 0<x<2 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Case 2: |x| > a x > a or Solve and graph |x| > 6 x>6 OR x < -a Read as “The distance of x from zero is greater than 6” x<-6 Graph -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Solve and graph |x – 3| ≥ 1 Read as “The distance of x - 3 from zero is greater than or equal to 1” x– 3 ≥ 1 +3 +3 x ≥ 4 OR x– 3 ≤-1 +3 +3 x ≤ 2 Graph -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Solve and graph |x + 3| + 2 > 5 |x + 3 | + 2 > 5 Before start solving, you MUST ISOLATE the absolute value expression on one side -2 -2 |x + 3 | > 3 x+3>3 -3 -3 x > 0 OR x+3 <-3 -3 -3 x < -6 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Special cases 1: |x| < - 2 No Solution, because the distance can’t be less than zero 2: |x| > -2 Solution is all real numbers because the distance is always greater than negative

Home Work Page 142 #s, 4 -multiples of 4 -36, + 39, 40, 41, 45, 46