Absolute value word problems

These absolute value word problems in this lesson will explore real life situations that can be modeled by either an absolute value equation or an absolute value inequality.

You may need to review the lesson about how to solve absolute value equations and absolute value inequalities.

Problem #1:

Your have money in your wallet, but you don't know the exact amount. When a friend asks you, you say that you have 50 dollars give or take 15. Use an absolute value equation to find least and biggest amount of money in your pocket? 

Solution:

Let x be the possible amount of money in your pocket. 

|x - 50| =  15

Equation #1

x - 50 = 15

x - 50 + 50 = 15 + 50

x = 65

Equation #2

x - 50 = -15

x - 50 + 50 = -15 + 50

x = 35

The least amount is 35 dollars and the biggest amount is 65 dollars.

Problem #2:

The ideal diameter of a piece of metal rod is 2.50 inches with an allowable error of at most 0.05 inch. Which rod(s) will you pick?

A. 2.5167  inches   B.    2.4417 inches      C.  2.484  inches     D.  2.558 inches     

Solution:

Let x be the actual diameter

|x - 2.50| ≤ 0.05

Equation #1

x - 2.50 ≤ 0.05

x - 2.50 + 2.50 ≤ 0.05 + 2.50

x ≤ 2.55

Equation #2

- (x - 2.50) ≤ 0.05

x - 2.50 ≥ - 0.05

x - 2.50 + 2.50 ≥ -0.05 + 2.50

x ≥ 2.45

2.45 ≤ x

Putting it all together, we get 2.45 ≤ x ≤ 2.55

We can pick A or C

       
Tricky absolute value word problems

Problem #3:

The ideal selling price of a Toyota is 25000. The dealer allows this price to vary 5%. What is the lowest price this dealer can sell this Toyota??

Solution:

The little trick is to remember to take 5% of 25000.

|x - 25000| ≤ 1250

Solving -(x - 25000) ≤ 1250 will give us our answer

x - 25000 ≥ -1250

x - 25000 + 25000 ≥ -1250 + 25000

x ≥ 23750

The lowest price 23750 dollars

Problem #4:

You personal trainer tells you that your weight loss should be between 35 and 45 pounds to win a free training session. Write an absolute value inequality that model your weight loss.

Solution:

The trick is to understand the meaning of absolute value in terms of distance .

|x - a|  = d

What does the absolute value equality above mean? It means to find 2 numbers that are located at the same distance d from a.

For example, |x - 1| = 3

This means to find two numbers that are 3 units away from 1. These 2 numbers are -2 and 4 of course since 4 - 1 = 3 and 1 - -2 = 3

If you are given -2 and 4, you will need to work backward to find 1 and 3.

The number that  -2 and 4  are the same distance from is the number between -2 and 4. To find the number that is between -2 and 4, taking the average of -2 and 4 will suffice. 

To find the distance, just do 4 - 1 = 3

We can do the same thing to find the absolute value inequality for 35 and 45

Since 45 - 40 = 5, the distance is 5.

The inequality is |x - 40| < 5

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