## Word Problems with Absolute Value Inequalities - Expii

Word problems with absolute value inequalities, explanations (3).  ## (Video) Absolute Value Inequality Word Problems Here's a really good video by I ♥ π . She shows you how to systematically work through absolute inequality word problems.

To start off, she looks at translating word problems into mathematical expressions . A lot of absolute value inequality word problems have to do with the margin of error . In these, you want the difference between an unknown value and a goal value (that you know) to be less than a number. Watch the beginning of the video to find out how to set up that equation, then try the following problem.

It takes Jamil 40 minutes to get to work, plus or minus at most 8 minutes depending on traffic. If she leaves her house at 10:02, when's earliest she can expect to get in?

First, show find the range of times that it could take Jamil to get to work using an absolute value equation. Which of the following equations would work?

## Related Lessons

To be able to solve Word Problems with Absolute Value Inequalities, you first need to be able to translate word problems into mathematical expressions . It would also be helpful to understand how to solve inequalities , in general. It may be helpful to to review solving word problems with inequalities .

Word problems with absolute value inequalities often talk about absolute error or tolerance .

If you see any of these phrases, that's a good indicator that you need to translate the word problem into an inequality using absolute values.

Let's look at an example: Image source: by Hannah Bonville

We know that the error (the difference between the actual height of the chair and the target height of 80 cm) cannot be any more than 2.5 cm (which means the same thing as less than or equal to ).

We can write the inequality like this:

Recall that if our absolute value inequality uses < or ≤, we need to set up one inequality to solve.

What inequality represents this information?

−2.5≤x−80≤2.5

−5≤x−80≤2.5 When given a word problem relating to absolute value inequalities, first translate the words to math terms. In other words, represent the word problem as a mathematical equation or expression, and use variables to stand in for unknown quantities. Then, simply solve for the variables to solve the word problem!

Absolute value word problems often have phrases such as "give or take x units" or "margin of error" or "plus or minus x units" to let you know that the answers can be above OR below a given number.

Here is a graphic with an example! Image source: By Caroline Kulczycky

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Absolute value word problems

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?

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

|x - 50| =  15

Equation #1

x - 50 = 15

x - 50 + 50 = 15 + 50

Equation #2

x - 50 = -15

x - 50 + 50 = -15 + 50

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

## 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??

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

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.

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

Solving absolute value inequalities

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