## Absolute value equations

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The Absolute value equations exercise appears under the Algebra I Math Mission and Mathematics II Math Mission . This exercise practices solving absolute value and recognizing how these are different from but related to linear equations.

## Types of Problems [ ]

There is one type of problem in this exercise:

Solve the absolute value equation

## Strategies [ ]

Knowledge of the properties of equality and some experience with absolute value would help when performing this exercise.

- Once the absolute value is isolated, the absolute value can you be though of under it's distance definition to increase efficiency.
- Treat the absolute value as a variable, and isolate the absolute value first. Then the absolve value equation can be split into a "positive" equation and a "negative equation."
- If the absolute value expression is equal to a negative number there is no solution.

## Real-life Applications [ ]

- Absolute value can be used in applications connecting algebra to geometry.
- The absolute value function is an example of a continuous but non-differentiable function (specifically at zero).
- Absolute value extends into the concept of modulus for complex numbers.
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## 1.2: Absolute Value Equations

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- Darlene Diaz
- Santiago Canyon College via ASCCC Open Educational Resources Initiative

When solving equations with absolute value, the solution may result in more than one possible answer because, recall, absolute value is just distance from zero. Since the integer \(−4\) has distance \(4\) units from zero, and 4 has distance 4 units from zero, then there are two integers that have distance \(4\) from zero, \(−4,\: 4\). We extend this concept to algebraic absolute value equations. This is illustrated in the following example.

## Example \(\PageIndex{1}\)

Solve for \(x\): \(|x|=7\)

\[\begin{array}{rl}|x|=7&\text{Expression in the absolute value can be positive or negative} \\ x=7\text{ or }x=-7&\text{Solution}\end{array}\nonumber\]

Let’s think about the solution set. The equation is asking for all numbers in which the distance from zero is \(7\). Well, there are two integers that have a distance \(7\) from zero, \(−7\) and \(7\). Hence, the solution set \(\{−7, 7\}\).

The first set of rules for working with negative numbers came from \(7^{\text{th}}\) century India. However, in 1758, more than a thousand years later, British mathematician Francis Maseres claimed that negatives “Darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple.”

## Definition: Absolute Value

Absolute value for linear equations in one variable is given by \[\text{If }|x|=a,\text{ then }x=a\text{ or }x=-a\nonumber\] where \(a\) is a real number.

When we have an equation with absolute value, it is important to first isolate the absolute value, then remove the absolute value by applying the definition.

## Example \(\PageIndex{2}\)

Solve for \(x\): \(5+|x|=8\)

\[\begin{array}{rl}5+|x|=8&\text{Isolate the absolute value by subtracting }5\text{ from each side} \\ |x|=3&\text{Rewrite as two linear equations} \\ x=3\text{ or }x=-3&\text{Solution}\end{array}\nonumber\]

Thus, the solution set is \(\{-3,3\}\).

## Example \(\PageIndex{3}\)

Solve for \(x\): \(-4|x|=-20\)

\[\begin{array}{rl}-4|x|=-20&\text{Isolate the absolute value} \\ \frac{-4|x|}{-4}=\frac{-20}{-4}&\text{Divide each side by }-4 \\ |x|=5&\text{Rewrite as two linear equations} \\ x=5\text{ or }x=-5&\text{Solution}\end{array}\nonumber\]

Thus, the solution set is \(\{-5,5\}\).

Never combine the inside of the absolute value with factors or terms from outside the absolute value. We always have to isolate the absolute value first, then apply the definition to obtain two equations without the absolute value.

## Example \(\PageIndex{4}\)

Solve for \(y\): \(5|y| − 4 = 26\)

\[\begin{array}{rl}5|y|-4=26&\text{Isolate the absolute value term by adding }4\text{ to each side} \\ 5|y|=30&\text{Divide each side by }5 \\ \\ |y|=6&\text{Rewrite as two linear equations} \\ y=6\quad\text{or}\quad y=-6&\text{Solution}\end{array}\nonumber\]

Thus, the solution set is \(\{-6,6\}\).

## Absolute Value Equations with Different Solutions

Often, we will have linear arguments inside the absolute value which changes the solution. Previously, all solution sets have been opposite integers, but in these cases, the solution sets contain different sized numbers.

## Example \(\PageIndex{5}\)

Solve for \(t\): \(|2t − 1| = 7\)

\[\begin{array}{rl} |2t-1|=7&\text{The absolute value term is isolated. Rewrite as two linear equations.} \\ 2t-1=7\quad\text{or}\quad 2t-1=-7&\text{Solve each equation.}\end{array}\nonumber\]

Notice we have two equations to solve where each equation results in a different solution. In any case, we solve as usual.

\[\begin{array}{lll} 2t-1=7&& 2t-1=-7 \\ 2t=8&\text{or}& 2t=-6 \\ t=4&&t=-3\end{array}\nonumber\]

Thus, the solution set is \(\{-3,4\}\).

## Multiple-Step Absolute Value Equations

Example \(\pageindex{6}\).

Solve for \(x\): \(2 − 4|2x + 3| = −18\)

To isolate the absolute value, we first apply the addition rule for equations. Then apply the multiplication rule for equations.

\[\begin{array}{rl}2-4|2x+3|=-18&\text{Isolate the absolute value term by subtracting }2\text{ from each side} \\ -4|2x+3|=-20&\text{Divide each side by }-4 \\ |2x+3|=5&\text{Rewrite as two linear equations.} \\ 2x+3=5\quad\text{or}\quad 2x+3=-5\end{array}\nonumber\]

Solve each equation.

\[\begin{array}{lll}2x+3=5&&2x+3=-5 \\ 2x=2&\text{or}& 2x=-8 \\ x=1&&x=-4\end{array}\nonumber\]

We now have obtained two solutions, \(x = 1\) and \(x = −4\). Thus, the solution set is \(\{−4, 1\}\).

## Equations with Two Absolute Values

In this case, we have an absolute value on each side of the equals sign. However, even though there are two absolute values, we apply the same process. Recall, methods never change, only problems.

## Example \(\PageIndex{7}\)

Solve for \(m\): \(|2m-7|=|4m+6|\)

In order to apply the definition, we rewrite this equation as two linear equations, but with the left side as its positive and negative value: \[\begin{array}{rl}|2m-7|=|4m+6|&\text{Rewrite as two linear equations} \\ 2m-7=\color{blue}{4m+6}\color{black}{}\quad\text{or}\quad 2m-7=\color{blue}{-(4m+6)}\color{black}{}\end{array}\nonumber\]

Now, we can solve as usual. Be sure to distribute the negative for the equation on the right.

\[\begin{array}{lll} &&2m-7=\color{blue}{-}\color{black}{}(4m+6) \\ 2m-7=4m+6&&2m-7=\color{blue}{-}\color{black}{}4m\color{blue}{-}\color{black}{}6 \\ -13=2m&&6m-7=-6 \\ -\frac{13}{2}=m&\text{or}&6m=1 \\ &&m=\frac{1}{6}\end{array}\nonumber\]

Thus gives the solutions, \(m=-\frac{13}{2}\) or \(m=\frac{1}{6}\). Thus, the solution set is \(\left\{-\frac{13}{2},\frac{1}{6}\right\}\).

In Example \(\PageIndex{7}\) , because there are absolute value expressions on both sides of the equation, we could have easily applied the definition to the left side and obtained \[\color{blue}{2m-7}\color{black}{}=4m-6\quad\text{or}\quad\color{blue}{-(2m-7)}\color{black}{}=4m-6\nonumber\]

Then solved each linear equation as usual and obtained the same results.

## Special Cases

As we are solving absolute value equations, it is important to be aware of special cases. Remember, the result after evaluating absolute value must always be non-negative.

## Example \(\PageIndex{8}\)

Solve for \(x\): \(7 + |2x − 5| = 4\)

\[\begin{array}{rl}7+|2x-5|=4&\text{Isolate the absolute value term by subtracting }7\text{ from each side} \\ |2x-5|=-3&X\text{ False}\end{array}\nonumber\]

Careful! Observe the absolute value of \(2x − 5\) is a negative number. This is impossible with absolute value because the result after evaluating absolute value must always be non-negative . Thus, we say this equation has no solution .

## Absolute Value Equations Homework

Exercise \(\pageindex{1}\).

\(|x| = 8\)

## Exercise \(\PageIndex{2}\)

\(|b| = 1\)

## Exercise \(\PageIndex{3}\)

\(|5 + 8a| = 53\)

## Exercise \(\PageIndex{4}\)

\(|3k + 8| = 2\)

## Exercise \(\PageIndex{5}\)

\(|9 + 7x| = 30\)

## Exercise \(\PageIndex{6}\)

\(|8 + 6m| = 50\)

## Exercise \(\PageIndex{7}\)

\(|6 − 2x| = 24\)

## Exercise \(\PageIndex{8}\)

\(−7| − 3 − 3r| = −21\)

## Exercise \(\PageIndex{9}\)

\(7| − 7x − 3| = 21\)

## Exercise \(\PageIndex{10}\)

\(\frac{|−4b − 10|}{8} = 3\)

## Exercise \(\PageIndex{11}\)

\(8|x + 7| − 3 = 5\)

## Exercise \(\PageIndex{12}\)

\(5|3 + 7m| + 1 = 51\)

## Exercise \(\PageIndex{13}\)

\(3 + 5|8 − 2x| = 63\)

## Exercise \(\PageIndex{14}\)

\(|6b − 2| + 10 = 44\)

## Exercise \(\PageIndex{15}\)

\(−7 + 8| − 7x − 3| = 73\)

## Exercise \(\PageIndex{16}\)

\(|5x + 3| = |2x − 1|\)

## Exercise \(\PageIndex{17}\)

\(|3x − 4| = |2x + 3|\)

## Exercise \(\PageIndex{18}\)

\(\left|\frac{4x-2}{5}\right|=\left|\frac{6x+3}{2}\right|\)

## Exercise \(\PageIndex{19}\)

\(|n| = 7\)

## Exercise \(\PageIndex{20}\)

\(|x| = 2\)

## Exercise \(\PageIndex{21}\)

\(|9n + 8| = 46\)

## Exercise \(\PageIndex{22}\)

\(|3 − x| = 6\)

## Exercise \(\PageIndex{23}\)

\(|5n + 7| = 23\)

## Exercise \(\PageIndex{24}\)

\(|9p + 6| = 3\)

## Exercise \(\PageIndex{25}\)

\(|3n − 2| = 7\)

## Exercise \(\PageIndex{26}\)

\(|2 + 2b| + 1 = 3\)

## Exercise \(\PageIndex{27}\)

\(\frac{|-4-3n|}{4}=2\)

## Exercise \(\PageIndex{28}\)

\(8|5p + 8| − 5 = 11\)

## Exercise \(\PageIndex{29}\)

\(3 − |6n + 7| = −40\)

## Exercise \(\PageIndex{30}\)

\(4|r + 7| + 3 = 59\)

## Exercise \(\PageIndex{31}\)

\(5 + 8| − 10n − 2| = 101\)

## Exercise \(\PageIndex{32}\)

\(7|10v − 2| − 9 = 5\)

## Exercise \(\PageIndex{33}\)

\(8|3 − 3n| − 5 = 91\)

## Exercise \(\PageIndex{34}\)

\(|2 + 3x| = |4 − 2x|\)

## Exercise \(\PageIndex{35}\)

\(\left|\frac{2x-5}{3}\right|=\left|\frac{3x+4}{2}\right|\)

## Exercise \(\PageIndex{36}\)

\(\frac{|-n+6|}{6}=0\)

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## Algebra 1 : How to solve absolute value equations

Study concepts, example questions & explanations for algebra 1, all algebra 1 resources, example questions, example question #1 : how to solve absolute value equations.

Solve the absolute value equation:

Solve for x.

No solution

So therefore, the solution is x = –7, 1.

## Example Question #3 : How To Solve Absolute Value Equations

Find the solution to x for |x – 3| = 2.

|x – 3| = 2 means that it can be separated into x – 3 = 2 and x – 3 = –2.

So both x = 5 and x = 1 work.

x – 3 = 2 Add 3 to both sides to get x = 5

x – 3 = –2 Add 3 to both sides to get x = 1

## Example Question #4 : How To Solve Absolute Value Equations

Solve for x:

Because of the absolute value signs,

Subtract 2 from both sides of both equations:

## Example Question #5 : How To Solve Absolute Value Equations

There are two answers to this problem:

## Example Question #6 : How To Solve Absolute Value Equations

## Example Question #7 : How To Solve Absolute Value Equations

## Example Question #8 : How To Solve Absolute Value Equations

What is the solution set of this equation?

## Example Question #9 : How To Solve Absolute Value Equations

## Example Question #10 : How To Solve Absolute Value Equations

Solve each separately to obtain the solution set:

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## How to Solve Absolute Value Equations

## When Do You Flip the Inequality Sign?

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.

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:

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.

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 :

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:

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

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

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:

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.

Subtract 3 from both sides to get:

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!

## Related Articles

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- Khan Academy: Intro to absolute value equations and graphs
- Lamar University: Absolute Value Equations
- Wolfram MathWorld: Absolute Value

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## 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

## Ultimate Math Solver (Free) Free Algebra Solver ... type anything in there!

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## Absolute Value Equation Calculator

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## Example (Click to try)

About absolute value equations.

- Get the absolve value expression by itself.
- Set up two equations and solve them separately.

## Absolute Value Equation Video Lesson

- Khan Academy Video: Absolute Value Equations

Need more problem types? Try MathPapa Algebra Calculator

Clear Absolute Value Equation Calculator »

## IMAGES

## VIDEO

## COMMENTS

To solve absolute value equations, find x values that make the expression inside the absolute value positive or negative the constant. To graph absolute value functions, plot two lines for the positive and negative cases that meet at the expression's zero. The graph is v-shaped. Created by Sal Khan and CK-12 Foundation.

Unit 1 Introduction to algebra. Unit 2 Solving basic equations & inequalities (one variable, linear) Unit 3 Linear equations, functions, & graphs. Unit 4 Sequences. Unit 5 System of equations. Unit 6 Two-variable inequalities. Unit 7 Functions. Unit 8 Absolute value equations, functions, & inequalities. Unit 9 Quadratic equations & functions.

Lesson 1: Solving absolute value equations. Intro to absolute value equations and graphs. Solving absolute value equations. Absolute value equations. Worked example: absolute value equation with two solutions. Worked example: absolute value equations with one solution. Worked example: absolute value equations with no solution.

Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/algebra-home/alg-absolute-value...

The standard absolute value graph y=|x| has its vertex at (0, 0). If you want to change the point to be at (3,0), that means you are making x=3. Notice, these are on opposite sides of the "=". if you need to place them on the same side of the "=", then you would have x-3=0.

Example of solving an absolute value equationWatch the next lesson: https://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/absolu...

3 x x + 2 = 2. To solve the equation above, we first. multiply/divide. both sides of the equation by x + 2 , then solve the resulting. linear/quadratic. equation. Because the denominator of the rational expression is x + 2 , the only value of x that would lead to division by 0 is .

Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/algebra2/functions_and_graphs/piecewise-functions-tutorial/e/gra...

Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-neg...

The absolute value function is commonly used to measure distances between points. Applied problems, such as ranges of possible values, can also be solved using the absolute value function. The graph of the absolute value function resembles a letter V. It has a corner point at which the graph changes direction.

Solving Absolute Value Equations. Solving absolute value equations is as easy as working with regular linear equations. The only additional key step that you need to remember is to separate the original absolute value equation into two parts: positive and negative (±) components.Below is the general approach on how to break them down into two equations:

This page titled 1.2: Solving Absolute Value Equations is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

The Absolute value equations exercise appears under the Algebra I Math Mission and Mathematics II Math Mission. This exercise practices solving absolute value and recognizing how these are different from but related to linear equations. There is one type of problem in this exercise: Solve the absolute value equation: This problem provides an absolute value equation and asks the student to find ...

To isolate the absolute value, we first apply the addition rule for equations. Then apply the multiplication rule for equations. 2 − 4 | 2x + 3 | = − 18 Isolate the absolute value term by subtracting 2 from each side − 4 | 2x + 3 | = − 20 Divide each side by − 4 | 2x + 3 | = 5 Rewrite as two linear equations. 2x + 3 = 5 or 2x + 3 = − 5.

No solution. Correct answer: x = -7, 1. Explanation: First, split |x + 3| = 4 into two possible scenarios according to the absolute value. x + 3 = 4. x + 3 = −4. Looking at x + 3 = 4, we can solve for x by subtracting 3 from both sides, so that we get x = 1. Looking at x + 3 = −4, we can solve for x by subtracting 3 from both sides, so ...

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.

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.

Solve an absolute value equation using the following steps: Get the absolve value expression by itself. Set up two equations and solve them separately. Absolute Value Equation Video Lesson. Khan Academy Video: Absolute Value Equations; Need more problem types? Try MathPapa Algebra ...

Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/algebra-home/alg-absolute-value...

Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/algebra-home/alg-absolute-valu...