An absolute value inequality also has 3 parts:A variableA middle value (mean) A distance from the middle to the either end (must be the same)However you also need to decide if it is < or >

The equation you use is | variable – mean | < or > distanceOne possible question is that you are given the two answers for example the solutions are -7 < x < 7:

To write and absolute value from this you still need to find the middle(average of these two answers) 7 + - 7 divided by 2 = 0So 0 is your mean.Now you still need to find out how far it is from the middle to the end(answer). 7 – 0 = 7. This one was easy because the middle was 0. Since the answer is in the middle it is an and

problem so you use <. The equation you use is | x –0 | < 7 or |x| < 7

Example 1:

Since this is an or problem and graphed away from each other, The equation you use is | variable – mean | > distance

Write an absolute value equation that has x < -1 or x >11 as solutions.

To write and absolute value from this you still need to find the middle(average of these two answers) -1 + 11 divided by 2 = 5So 5 is your mean.Now you still need to find out how far it is from the middle to the end(answer). 11 – 5 = 6.

The equation is: | x – 5 | > 6

Practice 1: -2 ≤ z ≤ 4

EXAMPLE 2 Solve an inequality of the form |ax + b| ≤ c

A professional baseball should weigh 5.125 ounces, with a tolerance of 0.125 ounce. Write and solve an absolute value inequality that describes the acceptable weights for a baseball.

Baseball

SOLUTION

Write a verbal model. Then write an inequality.STEP 1

EXAMPLE 3 Solve an inequality of the form |ax + b| ≤ c

STEP 2 Solve the inequality.

Write inequality.

Write equivalent compound inequality.

Add 5.125 to each expression.

|w – 5.125| ≤ 0.125

– 0.125 ≤ w – 5.125 ≤ 0.125

5 ≤ w ≤ 5.25

So, a baseball should weigh between 5 ounces and 5.25 ounces, inclusive. The graph is shown below.

ANSWER

EXAMPLE 6

The thickness of the mats used in the rings, parallel bars, and vault events must be between 7.5 inches and 8.25 inches, inclusive. Write an absolute value inequality describing the acceptable mat thicknesses.

Gymnastics

SOLUTION

STEP 1 Calculate the mean of the extreme mat thicknesses.

Write a range as an absolute value inequality

EXAMPLE 6

Mean of extremes = = 7.875 7.5 + 8.25 2

Find the tolerance by subtracting the mean from the upper extreme.

STEP 2

Tolerance = 8.25 – 7.875

Write a range as an absolute value inequality

= 0.375

EXAMPLE 6

STEP 3 Write a verbal model. Then write an inequality.

A mat is acceptable if its thickness t satisfies |t – 7.875| ≤ 0.375.

ANSWER

Write a range as an absolute value inequality

Practice 2:The outdoor temperature ranged between 370 F and 620 F in a 24 hour period. Let t represent the temperature during this time period.