Word Problems with Absolute Value Inequalities

(Video) Absolute Value Inequality Word Problems

by I ♥ π

Here's a really good video by I ♥ π. She shows you how to systematically work through absolute inequality word problems.

$\large{\text{Summary}}$

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.

$$|x - 40| \le 8$$

We're trying to find the difference between $x$ and $40$, so we have to subtract one from the other ($|40 - x| \le 8$ would have worked too). The difference can't be more than $8$ minutes, so we put our absolute value section on the left and use the less than or equal to symbol. Then, we can solve the equation to find the range of possible times! Learn more about solving absolute value equations here.

To find more practice on solving compound inequalities, check out this lesson.

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:

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:

$$|x - 80| \le 2.5$$

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

$$-2.5 \le x - 80 \le 2.5$$

All we need to do now is solve for $x$ by adding $80$ to each section of the compound inequality (the left side, the middle, and the right side).

$$77.5 \le x \le 82.5$$

This tells us that Rochelle's chairs can be between $77.5$ and $82.5\text{ cm}$ in height, inclusive.

Word Problems with Absolute Value Inequalities

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!