# Absolute value equations

Solving absolute value equations is based on the idea that absolute value represents the distance between a point on the number line and zero. In this lesson, we will look at a few examples to understand how to solve these equations and also take a bit of a look at this idea of distance as it relates to solving absolute value equations.

## Steps for solving absolute value equations

When given an absolute value equation, you will first need to isolate the absolute value part of the equation. Then, you will write two equations based on the definition of absolute value (though sometimes, there will end up only being one equation). This sounds complicated, but it is only a step or two more than solving the typical linear equation.

Let’s try these steps out with some examples!

## Examples

In this first example, the absolute value part of the equation is already isolated, so only step two will apply. Whether or not this first step applies or not, you will always have zero, one, or two solutions to any absolute value equation.

### Example

Solve the equation:
$$|5x – 2| = 13$$

### Solution

As mentioned, the absolute value part is already isolated. Therefore, we will solve two equations without the absolute value: one where the 13 is positive and one where 13 is negative.

Equation 1:

\begin{align}5x – 2 &= 13\\ 5x &= 15\\ x &= 3\end{align}

Equation 2:

\begin{align}5x – 2 &= -13\\ 5x &= -11 \\ x &= -\dfrac{11}{5}\end{align}

So, there are two solutions to this equation:
$$x = \bbox[border: 1px solid black; padding: 2px]{3, -\dfrac{11}{5}}$$

In this next example, there will be a little more work since the absolute value part of the equation is not isolated. In this situation, you will always need to isolate this term before you write your two equations, or you will end up with incorrect answers.

### Example

Solve the equation:
$$4 + 3|x – 5| = 16$$

### Solution

Your first step here is to use algebra to isolate the absolute value part of the equation.

$$4 + 3|x – 5| = 16$$

Subtract 4 from both sides.

$$3|x – 5| = 12$$

Divide both sides by 3.

$$|x – 5| = 4$$

Now you can write and solve two equations, one where the 4 is negative and one where the 4 is positive. Remember to drop the absolute value symbol at this step.

Equation 1:

\begin{align}x – 5 &= 4\\ x &= 9\end{align}

Equation 2:

\begin{align}x – 5 &= -4\\ x &= 1\end{align}

Once again, there are two solutions to the equation:
$$x = \bbox[border: 1px solid black; padding: 2px]{1, 9}$$

## Absolute value equations with one solution or no solutions

In both of our examples above, there were two solutions so you may think that this is always the case. While this is often right, there are cases where there is only one solution and even when there are none. The next two examples will show when this happens.

### Example – one solution

Solve the equation:
$$6|x – 2| – 1 = -1$$

### Solution

As usual, we will first isolate the absolute value equation.

$$6|x – 2| – 1 = -1$$

$$6|x – 2| = 0$$

Divide both sides by 6.

$$|x – 2| = 0$$

Normally at this stage, we would write two equations without the absolute value bars, but writing 0 with a positive or negative is the same thing. So we only have one equation:

$$x – 2 = 0$$

Adding 2 to both sides then gives the only solution.

$$x = \bbox[border: 1px solid black; padding: 2px]{2}$$

We will look more closely at why this happens, but first let’s look at how you might end up with no solutions.

### Example – no solutions

Solve the equation:
$$|10x – 1| + 3 = -8$$

### Solution

To isolate the absolute value, subtract 3 from both sides.

$$|10x – 1| = -11$$

At this step, it can be determined that there are no solutions to the equation. Why? The absolute value of any number is positive. Here, we have the absolute value of something is negative. This is not possible so there are no possible x-values that make this equation true. Therefore, you can write:

The absolute value of any number is always positive. Use this to determine when there are no solutions to an absolute value equation.

Notice that in both examples, the steps were the same as before. You will always follow those two steps when solving any absolute value equation.

## Why does this work?

You can think of the absolute value of any number as representing how far it is from zero on the number line. Consider $$|3|$$ and $$|–3|$$ below.

This is why the absolute value is always positive – it is representing a distance. Now think of an equation where the absolute value part is isolated, such as $$|5x + 1| = 2$$. Is the absolute value is 2, then all you know is that $$5x + 1$$ is 2 units from zero on the number line. This gives two possibilities:

So this is why we end up with two different equations. In the case of only one solution, you end up with an absolute value expression equal to zero. Since this means that the distance from zero on the number line is zero, you end up with only one equation.

## Another perspective

When you study the graphs of absolute value equations, you can see the three cases of one solution, no solution, and two solutions graphically. This is due to the shape of the graph of the absolute value function. This is a bit beyond the scope of this lesson, but in the graph below, you can see the graph of $$y = |x – 1|$$ and $$y = 2$$. Notice that the two graphs intersect at two points. These represent the two solutions to the equation $$|x – 1| = 2$$.

You can probably see how a horizontal line might cross the graph at exactly one point (one solution) or at no points (no solutions). This would be just changing the number on the right hand side of the equation.

## Summary

Absolute value equations are always solved with the same steps: isolate the absolute value term and then write equations based on the definition of the absolute value. There may end up being two solutions, one solution, or no solutions. To catch when there is no solution, always remember that absolute values must be positive, but remember to apply this idea only after the absolute value term has been isolated.