How to Solve Absolute Value Equations

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Absolute value equations can be a little intimidating at first, but if you keep at it you'll soon be solving them easily. When you're trying to solve absolute value equations, it helps to keep the meaning of absolute value in mind.

Definition of Absolute Value

The ​absolute value​ of a number ​x​, written | ​x​ |, is its distance from zero on a number line. For instance, −3 is 3 units away from zero, so the absolute value of −3 is 3. We write it like this: | −3 | = 3.

Another way to think about it is that ​absolute value​ is the positive "version" of a number. So the absolute value of −3 is 3, while the absolute value of 9, which is already positive, is 9.

Algebraically, we can write a ​formula for absolute value​ that looks like this:

| x | = \begin{cases} x &\text{if } x≥ 0 \\ -x &\text{if } x ≤ 0 \end{cases}

Take an example where ​x​ = 3. Since 3 ≥ 0, the absolute value of 3 is 3 (in absolute value notation, that's: | 3 | = 3).

Now what if ​x​ = −3? It's less than zero, so | −3 | = −( −3). The opposite, or "negative," of −3 is 3, so | −3 | = 3.

Solving Absolute Value Equations

Now for some absolute value equations. The general steps for solving an absolute value equation are:

Isolate the absolute value expression.

Solve the positive "version" of the equation.

Solve the negative "version" of the equation by multiplying the quantity on the other side of the equals sign by −1.

Take a look at the problem below for a concrete example of the steps.

Example: Solve the equation for ​x​:

| 3 + x | - 5 = 4

You'll need to get | 3 + ​x​ | by itself on the left side of the equals sign. To do this, add 5 to both sides:

| 3 + x | - 5 + 5 = 4 + 5 \\ | 3 + x | = 9

Solve for ​x​ as if the absolute value sign weren't there!

| 3 + x | = 9 → 3 + x = 9

That's easy: Just subtract 3 from both sides.

3 + x -3 = 9 -3 \\ x = 6

So one solution to the equation is that ​x​ = 6.

Start again at | 3 + ​x​ | = 9. The algebra in the previous step showed that ​x​ could be 6. But since this is an absolute value equation, there's another possibility to consider. In the equation above, the absolute value of "something" (3 + ​x​) equals 9. Sure, the absolute value of positive 9 equals 9, but there's another option here too! The absolute value of −9 also equals 9. So the unknown "something" could also equal −9.

In other words:

3 + x = -9

The quick way to arrive at this second version is to multiply the quantity on the other side of the equals from the absolute value expression (9, in this case) by −1, then solve the equation from there.

So:

| 3 + x | = 9 → 3 + x = 9 × ( -1) \\ 3 + x = -9

Subtract 3 from both sides to get:

3 + x -3 = -9 -3 \\ x = -12

So the two solutions are: ​x​ = 6 or ​x​ = −12.

And there you have it! These kinds of equations take practice, so don't worry if you're struggling at first. Keep at it and it will get easier!