4.5.1 – Solving Absolute Value Inequalities

• We’ve now addressed how to solve absolute value equations

• We can extend absolute value to inequalities• Remember, the absolute value equation y = |

x| is asking for the distance a number x is from zero (left or right)

Inequalities

• An absolute value inequality is asking for the values that will either be between certain numbers, or outside those numbers

• Two cases we will have to consider

Case 1

• When given the absolute value inequality |ax + b| > c OR |ax + b| ≥ c, we will setup 2

inequalities to solve

• 1) ax + b > c (or ≥)• OR• 2) ax + b < -c (or ≤)

• Want to go further away on the distance

• Example. Solve the absolute value inequality |x + 4| > 9

• Two inequalities?

• Example. Solve the absolute value inequality |2x – 5| ≥ 13

• Two inequalities?

Case 2

• The second case will involve staying between two values

• When given the absolute value inequality |ax + b| < c or |ax + b| ≤ c, we will set up the following inequality;

• -c < ax + b < c • -c ≤ ax + b ≤ c

• Example. Solve the absolute value inequality |x + 8| < 10

• Inequality?

• Example. Solve the absolute value inequality |-4 + 3x| ≤ 14

• Inequality?

Application

• Example. The absolute value inequality |t – 98.4| ≤ 0.6 is a model for normal body temperatures of humans at time t. Find the maximum and minimum the internal temperature of a body should be.

• Assignment• Pg. 201• 5-10, 21-29 odd, 34-38, 46, 48