how to solve absolute value equations

How to solve absolute value equations

|x + 5| = 3.

Worksheet on Abs Val Equations Abs Val Eqn Solver

The General Steps to solve an absolute value equation are:

• Rewrite the absolute value equation as two separate equations, one positive and the other negative
• Solve each equation separately
• After solving, substitute your answers back into original equation to verify that you solutions are valid
• Write out the final solution or graph it as needed

It's always easiest to understand a math concept by looking at some examples so, check outthe many examples and practice problems below.

You can always check your work with our Absolute value equations solver too

Practice Problems

Example equation.

Solve the equation: | X + 5| = 3

Click here to practice more problems like this one , questions that involve variables on 1 side of the equation.

Some absolute value equations have variables both sides of the equation. However, that will not change the steps we're going to follow to solve the problem as the example below shows:

Solve the equation: |3 X | = X − 21

Solve the following absolute value equation: | 5X +20| = 80

Solve the following absolute value equation: | X | + 3 = 2 X

This first set of problems involves absolute values with x on just 1 side of the equation (like problem 2 ).

Solve the following absolute value equation: |3 X −6 | = 21

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• \left|3x+1\right|=4
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• \left|3x+4\right|=-3
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Solving Simpler Absolute-Value Equations

Simpler Harder Special Case

When we take the absolute value of a number, we always end up with a positive number (or zero). Whether the input was positive or negative (or zero), the output is always positive (or zero). For instance, | 3 | = 3 , and | −3 | = 3 also.

This property — that both the positive and the negative become positive — makes solving absolute-value equations a little tricky. But once you learn the "trick", they're not so bad. Let's start with something simple:

Content Continues Below

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Solving Absolute Value Equations

Solve | x | = 3

I've pretty much already solved this, in my discussion above:

| −3 | = 3

So then x must be equal to 3 or equal to −3 .

But how am I supposed to solve this if I don't already know the answer? I will use the positive / negative property of the absolute value to split the equation into two cases, and I will use the fact that the "minus" sign in the negative case indicates "the opposite sign", not "a negative number".

For example, if I have x = −6 , then " − x " indicates "the opposite of x " or, in this case, −(−6) = +6 , a positive number. The "minus" sign in " − x " just indicates that I am changing the sign on x . It does not indicate a negative number. This distinction is crucial!

Whatever the value of x might be, taking the absolute value of x makes it positive. Since x might originally have been positive and might originally have been negative, I must acknowledge this fact when I remove the absolute-value bars. I do this by splitting the equation into two cases. For this exercise, these cases are as follows:

a. If the value of x was non-negative (that is, if it was positive or zero) to start with, then I can bring that value out of the absolute-value bars without changing its sign, giving me the equation x = 3 .

b. If the value of x was negative to start with, then I can bring that value out of the absolute-value bars by changing the sign on x , giving me the equation − x = 3 , which solves as x = −3 .

Then my solution is

x = ±3

We can, by the way, verify the above solution graphically. When we attempt to solve the absolute-value equation | x  | = 3 , we are, in effect, setting two line equations equal to each other and finding where they cross. For instance:

In the above, I've plotted the graph of y 1  = |  x  | (being the blue line that looks like a "V") and y 2  = 3 (being the green horizontal line). These two graphs cross at x  = −3 and at x  = +3 (being the two red dots).

If you're wanting to check your answers on a test (before you hand it in), it can be helpful to plug each side of the original absolute-value equation into your calculator as their own functions; then ask the calculator for the intersection points.

Of course, any solution can also be verified by plugging it back into the original exercise, and confirming that the left-hand side (LHS) of the equation simplifies to the same value as does the right-hand side (RHS) of the equation. For the equation above, here's my check:

x = −3

LHS: | x | = | −3 |

LHS: | x | = | +3 |

If you're ever in doubt about your solution to an equation, try graphing or else try plugging your solution back into the original question. Checking your work is always okay!

The step in the above, where the absolute-value equation was restated in two forms, one with a "plus" and one with a "minus", gives us a handy way to simplify things: When we have isolated the absolute value and go to take off the bars, we can split the equation into two cases; we will signify these cases by placing a "minus" on the opposite side of the equation (for one case) and a "plus" on the opposite side (for the other). Here's how this works:

Solve | x + 2 | = 7 , and check your solution(s).

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The absolute value is isolated on the left-hand side of the equation, so it's already set up for me to split the equation into two cases. To clear the absolute-value bars, I must split the equation into its two possible two cases, one each for if the contents of the absolute-value bars (that is, if the "argument" of the absolute value) is negative and if it's non-negative (that is, if it's positive or zero). To do this, I create two new equations, where the only difference between then is the sign on the right-hand side. First, I'll do the "minus" case:

x + 2 = −7

x = −9

Now I'll do the non-negative case, where I can just drop the bars and solve:

Now I need to check my solutions. I'll do this by plugging them back into the original equation, since the grader can't see me checking plots on my graphing calculator.

x = −9:

LHS: |(−9) + 2|

= |−7| = 7 = RHS

LHS: |(5) + 2|

= |7| = 7 = RHS

Both solutions check, so my answer is:

x = −9, 5

Solve | 2 x − 3 | − 4 = 3

First, I'll isolate the absolute-value part of the equation; that is, I'll get the absolute-value expression by itself on one side of the "equals" sign, with everything else on the other side:

| 2 x − 3 | − 4 = 3

| 2 x − 3 | = 7

Now I'll clear the absolute-value bars by splitting the equation into its two cases, one for each sign on the argument. First I'll do the negative case:

2 x − 3 = −7

2 x = −4

x = −2

And then I'll do the non-negative case:

2 x − 3 = 7

The exercise doesn't tell me to check, so I won't. (But, if I'd wanted to, I could have plugged "abs(2X−3)−4" and "3" into my calculator (as Y1 and Y2, respectively), and seen that the intersection points were at my x -values.) My answer is:

x = −2, 5

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How to Solve Absolute Value Equations

Last Updated: January 31, 2023 References

This article was co-authored by wikiHow Staff . Our trained team of editors and researchers validate articles for accuracy and comprehensiveness. wikiHow's Content Management Team carefully monitors the work from our editorial staff to ensure that each article is backed by trusted research and meets our high quality standards. This article has been viewed 23,117 times.

Setting up the Problem

• For example, |9| = 9; |-9| = -(-9) = 9.

Calculating the Values

Check Your Work

Expert Q&A

Video . by using this service, some information may be shared with youtube..

• Remember that absolute value bars are distinct from parentheses and function differently. Thanks Helpful 0 Not Helpful 0
• Once you've solved for any variables, remember to simplify absolute values accordingly. Thanks Helpful 0 Not Helpful 0

You Might Also Like

• ↑ http://tutorial.math.lamar.edu/Classes/Alg/SolveAbsValueEqns.aspx
• ↑ https://www.mathsisfun.com/numbers/absolute-value.html
• ↑ http://www.varsitytutors.com/high_school_math-help/solving-absolute-value-equations
• ↑ http://www.purplemath.com/modules/solveabs.htm
• ↑ https://www.khanacademy.org/math/algebra/absolute-value-equations-functions/absolute-value-equations/v/absolute-value-equations

About this article

To solve absolute value equations, first isolate the absolute value terms by moving anything outside of the vertical bars to the other side of the equation. Next, solve for the positive value of the equation by isolating the variable. Since the absolute variable can represent 2 numbers, then solve for the negative value by putting a negative sign outside the vertical bars. Then, move the negative by dividing both sides by -1 and solve for the variable. If you want to learn how to check your answers for an absolute value equation, keep reading the article! Did this summary help you? Yes No

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How to Solve Absolute Value Equations?

In this blog post, you will learn how to solve absolute value equations using a few simple steps.

The absolute value of a number is its distance from zero. The distance is always positive. For example \(2\) and \(-2\) have the same absolute value \(2\). The absolute value of \(a\) is written as \(|a|\). If \(a\) is positive, \(|a|\) is equal to \(a\). If \(a\) is negative, then the absolute value is its opposite: \(|a|=-a\)

Related Topics

• How to Solve an Absolute Value Inequality
• How to Graph Absolute Value Inequalities

A step-by-step guide to solving absolute value equations

To solve an absolute value equation, follow four steps:

• Step 1: Isolate the absolute value expression.
• Step 2: set its contents equal to both the positive and negative value of the number on the other side of the equation.
• Step 3: Solve both equations.
• Step 4: Check the solutions.

Solving Absolute value Equations – Example 1 :

Solve \(|x|-6=4\).

Add \(6\) to both sides of equation: \(|x|-6+6=4+6\) Then \(|x|=10\).

Set the contents of the absolute value portion equal to \(+10\) and \(-10\).

\(x=10\) or \(x=-10\)

Now, check the solutions:

\(|10|-6=10-6=4\) and \(|-10|-6=10-6=4\)

The answers are \(10\) and \(-10\).

Solving Absolute value Equations – Example 2:

Solve \(|x-3|=8\).

\(x-3=8\) or \(x-3=-8\)

Solve the equation \(x-3=8:\). add \(3\) to both sides: \(x-3+3=8+3\) then \(x=11\).

Solve the equation \(x-3=-8:\). add \(3\) to both sides: \(x-3+3=-8+3\) then \(x=-5\).

\(x=11\) → \(|11-3|=|8|=8\)

\(x=-5\) → \(|-5-3|=|-8|=8\)

The answers are \(11\) and \(-8\).

Exercises for Absolute Value Equations

Solve each absolute value equation..

• \(\color{blue}{|x|-2=5}\)
• \(\color{blue}{|x+3|=9}\)
• \(\color{blue}{|x-1|=3}\)
• \(\color{blue}{2|x+4|=12}\)
• \(\color{blue}{x=7}\)
• \(\color{blue}{x=6, x=-12}\)
• \(\color{blue}{x=4, x=-2}\)
• \(\color{blue}{x=2, x=-10}\)

by: Effortless Math Team about 2 years ago (category: Articles )

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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. Table of Contents

Steps for solving absolute value equations

• Step-by-step examples
• Absolute value equations with no solutions or one solution
• Why does our method work?

Another perspective

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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!

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.

Solve the equation: \(|5x – 2| = 13\)

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.

Solve the equation: \(4 + 3|x – 5| = 16\)

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.

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

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

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: \(6|x – 2| – 1 = -1\)

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

\(6|x – 2| – 1 = -1\)

Add 1 to both sides.

\(6|x – 2| = 0\)

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: \(|10x – 1| + 3 = -8\)

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:

Answer : No solutions

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.

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.

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.

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