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How to Solve Inequalities with Absolute Value on the ACT

As when solving an ACT Math problem that includes an expression with absolute value, you also need to split an inequality with absolute value into two separate inequalities. However, keep in mind one twist: One of the two resulting inequalities is simply the original inequality with the bars removed. The other inequality is the original inequality with

  • The bars removed

  • The opposite side negated (as with absolute value equations)

  • The inequality reversed (as with inequalities when you multiply or divide by a negative number)

These rules aren’t difficult, but they’re a little complicated, so be careful to do all three parts correctly.

Example 1

Which of the following values is in the solution set of

image0.png

(A)    0

(B)    2

(C)    –2

(D)    4

(E)    –4

Begin by splitting the inequality:

image1.png

Notice that the second of these two inequalities has the bars removed, the right side negated, and the inequality sign reversed. You’re now ready to solve both of these inequalities for t:

image2.png

To make these inequalities a little easier to read, put them in the following form:

image3.png

Thus, 0 falls into the range of solutions, so the right answer is Choice (A).

In some cases, the solution to an inequality with absolute value can lead to a pair of inequalities that appear to contradict each other. When this happens, both inequalities aren’t true, but at least one of them is, so link them with the word or. This concept is a little tricky, so don’t worry if it’s not making sense. The next problem provides a concrete example.

Example 2

What is the solution set for

image4.png

Before you begin, notice that the original inequality is

image5.png

so no solution can include either

image6.png

As a result, you can rule out Choices (G) and (J). Now isolate

image7.png

on the left side of the inequality:

image8.png

You’re now ready to remove the bars and split the inequality:

image9.png

Notice that the second of these two inequalities has the bars removed, the right side negated, and the inequality sign reversed. You’re now ready to solve the first one:

image10.png

Next, solve the second inequality:

image11.png

Notice that the two solutions

image12.png

seem to contradict each other: If n is greater than 4, how can it be less than 1? When this situation occurs, either solution can be true, so link the two resulting solutions with the word or:

image13.png

Thus, the correct answer is Choice (K).

Be extra careful when working with an inequality that sets an absolute value either greater than or greater than or equal to another value that includes a variable. This type of inequality can sometimes produce a false (or extraneous) solution — that is, a solution that appears correct but doesn’t work when plugged back into the problem. The next example shows you how and why this can happen.

Which of the following is the solution set for

image14.png

To begin, remove the absolute value bars, split the inequality, and solve each separately:

image15.png

According to this result, x < 1 and x < –3 both appear correct, so you may be tempted to choose Choice (E). However, if this answer were correct, then x = 0 should be outside the solution set. So plugging 0 into the original inequality should give you the wrong answer:

image16.png

This solution is unexpected. In fact, x = 0 is in the solution set for this inequality.

What went wrong? Take another look at the original inequality:

image17.png

This inequality sets an absolute value greater than 2x. So if x is any negative number, the absolute value (which can never be negative) must be in the solution set. Therefore, the solution x < –3 is false because it tells you that only certain negative values of x are in the solution set. Throwing out this false solution leaves you with the correct answer, which is x < 1; so the correct answer is Choice (A).

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