A -3 ≥ x ≥ 1 B -3 > x > 1 C -3 < x < 1 D -3 ≤ x ≤ 1 30 Sec Which compound inequality is shown by the graph? 1 –3 –2 –1 3 2 –4 4 5 6 –5 –6 A -3 ≥ x ≥ 1 B -3 > x > 1 C -3 < x < 1 D -3 ≤ x ≤ 1

A x ≥ 0 or x ≤ -5 B x > 0 or x ≤ -5 C x ≤ 0 or x ≥ -5 30 Sec Which compound inequality is shown by the graph? 1 –3 –2 –1 3 2 –4 4 5 6 –5 –6 A x ≥ 0 or x ≤ -5 B x > 0 or x ≤ -5 C x ≤ 0 or x ≥ -5 D x < 0 or x ≤ -5

Solving absolute Value inequality Lesson 2-7 Solving absolute Value inequality You'll Learn how to Solve absolute value inequality

Absolute value The absolute value of a number is the distance of that number from zero EXAMPLE 1 | x | = 7 Read as “The distance of my house from the city center is 7” = 7 or = -7 1 2 3 4 5 6 7 -1 -2 -3 -4 -5 -6 -7 center EXAMPLE 2 Solve and graph |x + 4| = 2 x + 4 = 2 or x + 4 = -2 x = -2 x = -6 1 2 3 4 5 6 7 -1 -2 -3 -4 -5 -6 -7

3 4 Case1 “and” | x | < 5 < 5 and > -5 Solve the inequality EXAMPLE 3 | x | < 5 Read as “The distance of my house from the city center is < 5” < 5 and > -5 1 2 3 4 5 6 7 -1 -2 -3 -4 -5 -6 -7 center EXAMPLE 4 Solve the inequality |x| – 3 < – 2 ISOLATE the absolute value expression |x| < 1 Write as compound inequality x < 1 and x > -1 1 2 3 4 5 6 7 -1 -2 -3 -4 -5 -6 -7

5 6 Case2 “OR” | x | > 3 > 3 or < -3 Solve the inequality EXAMPLE 5 | x | > 3 Read as “The distance of my house from the city center is > 3” > 3 or < -3 1 2 3 4 5 6 7 -1 -2 -3 -4 -5 -6 -7 EXAMPLE 6 Solve the inequality |x| + 14 ≥ 19 ISOLATE the absolute value expression |x| ≥ 5 Write as compound inequality x ≥ 5 or x ≤ -5 1 2 3 4 5 6 7 -1 -2 -3 -4 -5 -6 -7

7 Special Cases: Solve and graph a) |x| + 5 < 2 |x| < -3 EXAMPLE 7 Solve and graph a) |x| + 5 < 2 ISOLATE the absolute value expression |x| < -3 Think: distance is never < zero Conclusion: No Solution b) |x| + 3 > 2 ISOLATE the absolute value expression |x| > -1 Think: distance is always > zero Conclusion: Solution is all real numbers Lesson Summary |x| < a (positive) x < a and x > -a |x| < a (negative) No Solution |x| > a (positive) x > a or x < -a |x| > a (negative) Solution is all real numbers

Solve and graph a) |x| – 3 < –1 b) |x – 1| ≤ 2 |x| < 2 x – 1 ≤ 2 Exercises Solve and graph a) |x| – 3 < –1 b) |x – 1| ≤ 2 ISOLATE |x| < 2 Split x – 1 ≤ 2 and x – 1 ≥ -2 x < 2 and x > -2 c) |x| + 12 > 13 d) 3 + |x + 2| > 5 ISOLATE |x| > 1 |x + 2| > 2 Split x > 1 or x < -1 x + 2 > 2 or x + 2 < -2 e) |x + 4| – 5 > –8 f) |x + 2| + 9 < 7 ISOLATE |x + 4| > -3 |x + 2| < -2 Solution is all real #s No Solution