Approaches to Solving Absolute Value Equations

A couple of things before we start this lesson. First, review the lesson on absolute values to make sure you've mastered that skill. Second, remember that absolute value can never be a negative number.

There are a few steps that you can consistently follow to solve equations that contain variables and absolute values:

1. Isolate the absolute value

2. Write two equations: one with a positive solution and one with a negative solution

3. Solve for $x$ in both equations

Let's take a look at an example of absolute value equations with sums to see these steps in action:

$$3|2x - 5| + 7 = 22$$

Step 1: Isolate the absolute value expression

We need to get the absolute value ($|2x-5|$) alone on one side of the equation.

$$\begin{align} 3|2x - 5| + 7 &= 22\\ 3|2x - 5| &= 15\\ |2x - 5| &= 5\\ \end{align}$$

Step 2: Write two equations: one with a positive solution and one with a negative solution

Remove the absolute value sign and set up two equations:

$$\begin{align} 2x - 5 &= 5\\ 2x - 5 &= -5 \end{align}$$

Step 3: Solve for $x$ in both equations

What are the two possible values of $x$?

$$\begin{align} 2x - 5 &= 5\\ 2x &= 10\\ x &= 5\\\ \\ OR\\\ \\ 2x - 5 &= -5\\ 2x &= 0\\ x &= 0 \end{align}$$

Notice how both values for $x$, when plugged into the original equation, produce the same result. This is a good way to check your answer.

We can also write the values of $x$ as a solution set that looks like this: $$\{0, 5\}$$

Way 1: Two Cases

You can solve absolute value equations breaking them into two cases. For example, suppose you're given the equation:

$$|x-5|=4$$.

Because of the way absolute values work, we can rewrite this as $x-5=\pm4$ and separate it into two equation. Both will produce a solution:

$$\begin{align} x-5=4\\ x-5=-4 \end{align}$$

These result in $x=9$ and $x=1$, which both satisfy the equation.

This works because an absolute value turns the expression inside it to a positive value. For example, both of the following equations are true:

$$\begin{align}|5|&=5\\ |-3|&=3\end{align}$$

(Surprisingly enough, you can use the two cases method to produce the general solution to the equation $|x+a|=b$, which is $x=\pm b-a$.)

Way 2: Number Lines and Definitions

You can also use a number line as well as the definition of absolute value to solve an absolute value equation. The definition, by the way, is that the absolute value of an expression is its distance from $0$ on the number line. Note that this method is very similar to the Two Cases method, but it's still different in its own way.

If you have an equation that looks like $|\text{[expression]}|=\text{[number]}$, you can plot points on the number line at $\text{[number]}$ and write $\text{[expression]}$ over them. (Don't literally write the word "expression", though!) From there, plot more points on the number line following the steps of simplifying $\text{[expression]}$ down to just your variable (for example, adding $2$, dividing by $7$, taking the square root, etc.), which will produce the solutions.

Way 2: Example

Let's say we have the absolute value equation

$$|2x - 5| = 3$$

Plot points on the number line at $3$ and $-3$ and write the expression $2x - 5$ over each one. I'm calling one point $A$ and one point $B$.

From there, plot two more points $5$ spaces to the right of the previous two and write $2x$ over each of those.

Finally, plot your last points: one at $\frac{A}{2}$ and $\frac{B}{2}$ and write $x$ over each one. This simulates the final step in the equation, dividing both sides by $2$. Those are your solutions!

When solving these types of equations, you may come across an equation with no solution.

Looking Further

Later in math, you will learn about absolute value functions, which are basically 2-dimensional versions of absolute value equations.

(Video) Solving Absolute Value Equations

by The Organic Chemistry Tutor

In this video, The Organic Chemistry Tutor goes over how to solve absolute value equations.

Summary

Absolute value is the distance from zero, so you cannot have a negative value. $$|5|-5$$ $$|-8|=8$$

So, if we're told that: $$|x|=7,$$ what are the possible values of $x$? Well, $x$ can equal $7$ or $-7$.

So how can we use this concept when solving absolute value equations? Say we have the equation, $|x+2|=9$. That means that: $$x+2=9$$ $$x+7$$ $$OR$$ $$x+2=-9$$ $$x=-11$$ Both of these equations are true.

Let's look at another example. Say $|2x-5|=1$. What are the possible values of $x$? Well, $|2x-5|$ can equal $+1$ or $-1$. $$2x-5=1$$ $$2x=6$$ $$x=3$$ $$OR$$ $$2x-5=-1$$ $$2x=4$$ $$x=2$$

What if we were given the equation: $$|5x-1|=-2$$ What are the values of $x$? Well, there is no solution to this equation, because the absolute value can never be negative.

Let's look at another example that's a bit more complex. What are the values of $x$ that satisfy: $3|x-1|-2=7$?

The first thing we want to do is isolate the absolute value part on one side of the equation. $$3|x-1|-2=7$$ $$3|x-1|=9$$ $$|x-1|=3$$ Then, we can solve it by setting up the absolute value part to equal $3$ or $-3$. $$x-1=3$$ $$x=4$$ $$OR$$ $$x-1=-3$$ $$x=-2$$