Absolute-Value Inequalities

Once you catch on to how these inequalities work, this stuff really isn't so bad.

(Note: This lesson covers linear absolute-value inequalities only.)

What are absolute values?

Recall the original definition of absolute values as distance: "| x | is the distance of x from zero." For instance, both −2 and +2 are two units from zero, as you can see in the image below:

This means that their absolute values will both be 2; that is, we have:

With this definition and picture in mind, let's look at some absolute value inequalities.

How do you solve "less than" absolute-value inequalities?

To solve a "less than" absolute-value inequality, we use the definition of the absolute value to restate the inquality as a three-part inequality; that is, given | mx + b | < c, convert this to:

Then solve the resulting three-part linear inequality to get the variable by itself in the middle.

• Solve | x | < 3, and graph its solution.

This pattern for "less than" absolute-value inequalities always holds:

By the way, the correct conjunction for "less than" absolute-value inequalities is "and". Why? Because the variable is contained within one interval. In the example above, x was both "more than −3" and also "less than +3". The x is in the interval that satisfied both inequalities at the same time. So "and" is the correct conjuction.

Even when the exercises get more complicated, the above pattern will still hold.

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• Solve | 2x + 3 | < 6.

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The other case for absolute value inequalities is the "greater than" case.

How do you solve "greater than" absolute-value inequalities?

To solve "greater than" absolute-value inequalities, use the definition of the absolute value to split the inequality into two cases; that is, for | mx + b | > c, split the inequality into its two cases:

Then solve the two linear inequalities separately.

• Solve | x | > 2, and graph.

Take careful note! The solution to this "greater than" absolute-value inequality is TWO regular inequalities, not one. DO NOT try to write this as one inequality. If you try to write this solution as "−2 > x > 2", your answer be counted wrong. Why? Because, if you take out the x in the middle, you'll see that you would be saying "−2 > 2", which certainly is not true. Take the extra half a second, and write the solution correctly.

This pattern for "greater than" absolute-value inequalities always holds:

And, by the way, the correct conjunction is "or", not "and". Why? Because the variable cannot be in both of the solution intervals and the same time. In the example above, x cannot be both "less than −2" and also "greater than +2" at the same time. Therefore, we use "or" for these types of solutions.

Even when the inequalities are more complicated, the above pattern still holds.

• Solve | 2x − 3 | > 5.

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What is the take-away for solving linear absolute-value inequalities?

1. Check the inequality symbol: is it "greater than" or "less than"?
2. If "less than", drop the absolute-value bars, restate as a three-part inequality, and solve with an "and" statement. Example: |x − 3| < 5 becomes −5 < (x − 3) < +5
3. If "greater than", drop the absolute-value bars, split the inequality into its two cases, and solve the two inequalities separately with an "or" statement. Example: |x − 3| > 5 becomes (x − 3) < −3 or (x − 3) > 3

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There is another situation you might encounter: You'll be given a pair of inequalities, and you'll be asked to find the corresponding absolute-value inequality. This process can feel a bit weird, so I'll give a couple examples of how it works.

• Find the absolute-value inequality statement that corresponds to −2 < x < 4.

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• Find the absolute-value inequality statement that corresponds to the inequalities x ≤ 19 or x ≥ 24

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Warning: There is one "trick" type of question for this kind of problem, where they'll try to trip you up on homework or tests. They'll ask you to solve something like "| x + 2 | < −1". But can an absolute value ever be negative, let alone be less than a negative? No! So there is no solution to this inequality; it doesn't even make sense. Don't waste a lot of time trying to "solve" this; just write down "no solution".

Similarly, if you're given something like "| x − 2 | > −3", the first thing to note is that all absolute values are zero or positive. In particular, they're never negative. They're asking you for the x-values that will make the absolute-value expression greater than a negative number. Since the absolute value will always be greater than any negative number, the solution must be "all x" or "all real numbers".