Solving absolute value inequalities is a lot like solving absolute value equations, but there are a couple of extra details to keep in mind. It helps to already be comfortable solving absolute value equations, but it's okay if you're learning them together too!
Definition of Absolute Value Inequality
First of all, an absolute value inequality is an inequality that involves an absolute value expression. For example,
| 5 + x | − 10 > 6 is an absolute value inequality because it has an inequality sign, >, and an absolute value expression, | 5 + x |.
How to Solve an Absolute Value Inequality
The steps to solving an absolute value inequality are much like the steps to solving an absolute value equation:
Step 1: Isolate the absolute value expression on one side of the inequality.
Step 2: Solve the positive "version" of the inequality.
Step 3: Solve the negative "version" of the inequality by multiplying the quantity on the other side of the inequality by −1 and flipping the inequality sign.
That's a lot to take in all at once, so here's an example that will walk you through the steps.
Solve the inequality for x: | 5 + 5_x_ | − 3 > 2.
Isolate the Absolute Value Expression
To do this, get | 5 + 5_x_ | by itself on the left side of the inequality. All you have to do is add 3 to each side:
| 5 + 5_x_ | − 3 (+ 3) > 2 (+ 3)
| 5 + 5_x_ | > 5.
Now there are two "versions" of the inequality that we need to solve: the positive "version" and the negative "version."
Solve the Positive "Version" of the Inequality
For this step, we'll assume that things are as they appear: that 5 + 5_x_ > 5.
| 5 + 5_x_ | > 5 → 5 + 5_x_ > 5.
This is a simple inequality; you just have to solve for x as usual. Subtract 5 from both sides, then divide both sides by 5.
5 + 5_x_ > 5
5 + 5_x_ (− 5) > 5 ( − 5) (subtract five from both sides)
5_x_ > 0
5_x_ (÷ 5) > 0 (÷ 5) (divide both sides by five)
x > 0.
Not bad! So one possible solution to our inequality is that x > 0. Now, since there are absolute values involved, it's time consider another possibility.
Solve the Negative "Version" of the Inequality
To understand this next bit, it helps to remember what absolute value means. Absolute value measures a number's distance from zero. Distance is always positive, so 9 is nine units away from zero, but −9 is also nine units away from zero.
So | 9 | = 9, but | −9 | = 9 as well.
Now back to the problem above. The work above showed that | 5 + 5_x_ | > 5; in other words, the absolute value of "something" is greater than five. Now, any positive number bigger than five is going to be further away from zero than five is. So the first option was that "something," 5 + 5_x_, is bigger than 5.
That is: 5 + 5_x_ > 5.
That's the scenario tackled above, in Step 2.
Now think a little further. What else is five units away from zero? Well, negative five is. And anything further along the number line from negative five is going to be even further away from zero. So our "something" could be a negative number that's further away from zero than negative five. That means it would be a bigger-sounding number, but technically less than negative five because it's moving in the negative direction on the number line.
So our "something," 5 + 5x, could be less than −5.
5 + 5_x_ < −5
The quick way to do this algebraically is to multiply the quantity on the other side of the inequality, 5, by negative one, then flip the inequality sign:
| 5 + 5x | > 5 → 5 + 5_x_ < − 5
Then solve as usual.
5 + 5_x_ < -5
5 + 5_x_ ( −5) < −5 ( − 5) (subtract 5 from both sides)
5_x_ < −10
5_x_ (÷ 5) < −10 (÷ 5)
x < −2.
So the two possible solutions to the inequality are x > 0 or x < −2. Check yourself by plugging in a few possible solutions to make sure the inequality still holds true.
Absolute Value Inequalities With No Solution
There is a scenario where there would be no solutions to an absolute value inequality. Since absolute values are always positive, they can't be equal to or less than negative numbers.
So | x | < −2 has no solution because the outcome of an absolute value expression has to be positive.
To write the solution to our main example in interval notation, think about how the solution looks on the number line. Our solution was x > 0 or x < −2. On a number line, that's an open dot at 0, with a line extending out to positive infinity, and an open dot at −2, with a line extending away to negative infinity. These solutions point away from each other, not toward each other, so take each piece separately.
For x > 0 on a number line, there's an open dot at zero and then a line extending out to infinity. In interval notation, an open dot is illustrated with parentheses, ( ), and a closed dot, or inequalities with ≥ or ≤, would use brackets, [ ]. So for x > 0, write (0, ∞).
The other half, x < −2, on a number line is an open dot at −2 and then an arrow extending all the way to −∞. In interval notation, that's (−∞, −2).
"Or" in interval notation is the union sign, ∪.
So the solution in interval notation is ( −∞, −2) ∪ (0, ∞).