2-7 absolute value inequalities Chapter 2 2-7 absolute value inequalities

Problem of the day

objectives Solve compound inequalities in one variable involving absolute-value expressions.

Absolute value inequalities When an inequality contains an absolute- value expression, it can be written as a compound inequality. The inequality |x| < 5 describes all real numbers whose distance from 0 is less than 5 units. The solutions are all numbers between –5 and 5, so |x|< 5 can be rewritten as –5 < x < 5, or as x > –5 AND x < 5.

Example#1 Solve the inequality and graph the solutions. |x|– 3 < –1 +3 +3 |x| < 2 |x|– 3 < –1 x > –2 AND x < 2 –2 –1 1 2 2 units

Example#2 Solve the inequality and graph the solutions. |x – 1| ≤ 2 x – 1 ≥ –2 AND x – 1 ≤ 2 +1 +1 +1 +1 x≥-1 AND x ≤ 3 –2 –1 1 2 3 –3

Example#3 Solve the inequality and graph the solutions. 2|x| ≤ 6 x ≥ –3 AND x ≤ 3 ___________ 2 2 2|x| ≤ 6 –2 –1 1 2 3 units –3 3

Example#4 Solve each inequality and graph the solutions.

Student guided practice Go to page 145 and do problems 1-6 from your book

Absolute value inequalities The inequality |x| > 5 describes all real numbers whose distance from 0 is greater than 5 units. The solutions are all numbers less than –5 or greater than 5. The inequality |x| > 5 can be rewritten as the compound inequality x < –5 OR x > 5.

Example#5 Solve the inequality and graph the solutions |x| + 14 ≥ 19

Example #6 Solve the inequality and graph the solutions.

Example#6 Solve each inequality and graph the solutions. |x| + 10 ≥ 12

Student guided practice Go to book page 145 and do problems 14-17

applications A dry-chemical fire extinguisher should be pressurized to 125 psi, but it is acceptable for the pressure to differ from this value by at most 75 psi. Write and solve an absolute- value inequality to find the range of acceptable pressures. Graph the solution.

Solution to problem p – 125 ≤ 75 |p – 125| ≤ 75 p – 125 ≥ –75 AND p – 125 ≤ 75 The range of pressure is 50 ≤ p ≤ 200. +125 +125 +125 +125 p ≥ 50 AND p ≤ 200

Absolute value inequalities When solving an absolute-value inequality, you may get a statement that is true for all values of the variable. In this case, all real numbers are solutions of the original inequality. If you get a false statement when solving an absolute-value inequality, the original inequality has no solutions.

Example#7 |x + 4|– 5 > – 8 All real numbers are solutions.

Example#8 |x – 2| + 9 < 7 The inequality has no solutions.

Student guided practice Go to your book page 145 and do problems 33-35

Homework!!! Do odd numbers from 20-31 from your book page 145

closure Today we saw about how to solve absolute value inequalities Next class we are going to continue with chapter 3