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Study Guides > College Algebra: Co-requisite Course

Equations and Inequalities with Absolute Value

Learning Objectives

  • Solve equations containing absolute values
  • Recognize when a linear equation that contains absolute value does not have a solution
  • Solve inequalities containing absolute values

Solving an Absolute Value Equation

Next, we will learn how to slve an absolute value equation. To solve an equation such as ∣2x−6∣=8|2x - 6|=8, we notice that the absolute value will be equal to 8 if the quantity inside the absolute value bars is 88 or −8-8. This leads to two different equations we can solve independently.
2x−6=8 or 2x−6=−82x=142x=−2x=7x=−1\begin{array}{lll}2x - 6=8\hfill & \text{ or }\hfill & 2x - 6=-8\hfill \\ 2x=14\hfill & \hfill & 2x=-2\hfill \\ x=7\hfill & \hfill & x=-1\hfill \end{array}
Knowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.

A General Note: Absolute Value Equations

The absolute value of x is written as ∣x∣|x|. It has the following properties:
If x≥0, then ∣x∣=x.If x<0, then ∣x∣=−x.\begin{array}{l}\text{If } x\ge 0,\text{ then }|x|=x.\hfill \\ \text{If }x<0,\text{ then }|x|=-x.\hfill \end{array}
For real numbers AA and BB, an equation of the form ∣A∣=B|A|=B, with B≥0B\ge 0, will have solutions when A=BA=B or A=−BA=-B. If B<0B<0, the equation ∣A∣=B|A|=B has no solution. An absolute value equation in the form ∣ax+b∣=c|ax+b|=c has the following properties:
If c<0,∣ax+b∣=c has no solution.If c=0,∣ax+b∣=c has one solution.If c>0,∣ax+b∣=c has two solutions.\begin{array}{l}\text{If }c<0,|ax+b|=c\text{ has no solution}.\hfill \\ \text{If }c=0,|ax+b|=c\text{ has one solution}.\hfill \\ \text{If }c>0,|ax+b|=c\text{ has two solutions}.\hfill \end{array}

How To: Given an absolute value equation, solve it.

  1. Isolate the absolute value expression on one side of the equal sign.
  2. If c>0c>0, write and solve two equations: ax+b=cax+b=c and ax+b=−cax+b=-c.
In the next video, we show examples of solving a simple absolute value equation. https://youtu.be/4g-o_-mAFpc

Example: Solving Absolute Value Equations

Solve the following absolute value equations:
  1. ∣6x+4∣=8|6x+4|=8
  2. ∣3x+4∣=−9|3x+4|=-9
  3. ∣3x−5∣−4=6|3x - 5|-4=6
  4. ∣−5x+10∣=0|-5x+10|=0

Answer: a. ∣6x+4∣=8|6x+4|=8 Write two equations and solve each:

6x+4=86x+4=−86x=46x=−12x=23x=−2\begin{array}{ll}6x+4\hfill&=8\hfill& 6x+4\hfill&=-8\hfill \\ 6x\hfill&=4\hfill& 6x\hfill&=-12\hfill \\ x\hfill&=\frac{2}{3}\hfill& x\hfill&=-2\hfill \end{array}

The two solutions are x=23x=\frac{2}{3}, x=−2x=-2. b. ∣3x+4∣=−9|3x+4|=-9 There is no solution as an absolute value cannot be negative. c. ∣3x−5∣−4=6|3x - 5|-4=6 Isolate the absolute value expression and then write two equations.
∣3x−5∣−4=6∣3x−5∣=103x−5=103x−5=−103x=153x=−5x=5x=−53\begin{array}{lll}\hfill & |3x - 5|-4=6\hfill & \hfill \\ \hfill & |3x - 5|=10\hfill & \hfill \\ \hfill & \hfill & \hfill \\ 3x - 5=10\hfill & \hfill & 3x - 5=-10\hfill \\ 3x=15\hfill & \hfill & 3x=-5\hfill \\ x=5\hfill & \hfill & x=-\frac{5}{3}\hfill \end{array}
There are two solutions: x=5x=5, x=−53x=-\frac{5}{3}. d. ∣−5x+10∣=0|-5x+10|=0 The equation is set equal to zero, so we have to write only one equation.
−5x+10=0−5x=−10x=2\begin{array}{l}-5x+10\hfill&=0\hfill \\ -5x\hfill&=-10\hfill \\ x\hfill&=2\hfill \end{array}
There is one solution: x=2x=2.

In the two videos that follow, we show examples of how to solve an absolute value equation that requires you to isolate the absolute value first using mathematical operations. https://youtu.be/-HrOMkIiSfU https://youtu.be/2bEA7HoDfpk

Try It

Solve the absolute value equation: ∣1−4x∣+8=13|1 - 4x|+8=13.

Answer: x=−1x=-1, x=32x=\frac{3}{2}

Absolute value equations with no solutions

As we are solving absolute value equations it is important to be aware of special cases. An absolute value is defined as the distance from 0 on a number line, so it must be a positive number. When an absolute value expression is equal to a negative number, we say the equation has no solution, or DNE. Notice how this happens in the next two examples.

Example

Solve for x. 7+∣2x−5∣=47+\left|2x-5\right|=4

Answer: Notice absolute value is not alone. Subtract 77 from each side to isolate the absolute value.

7+∣2x−5∣=4     −7                          −7 ‾∣2x−5∣=−3\begin{array}{r}7+\left|2x-5\right|=4\,\,\,\,\\\underline{\,-7\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-7\,}\\\left|2x-5\right|=-3\end{array}

Result of absolute value is negative! The result of an absolute value must always be positive, so we say there is no solution to this equation, or DNE.

Example

Solve for x. −12∣x+3∣=6-\frac{1}{2}\left|x+3\right|=6

Answer: Notice absolute value is not alone, multiply both sides by the reciprocal of −12-\frac{1}{2}, which is −2-2.

−12∣x+3∣=6                    (−2)−12∣x+3∣=(−2)6                     ∣x+3∣=−12     \begin{array}{r}-\frac{1}{2}\left|x+3\right|=6\,\,\,\,\,\,\,\,\,\,\,\,\\\,\,\,\,\,\,\,\,\left(-2\right)-\frac{1}{2}\left|x+3\right|=\left(-2\right)6\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left|x+3\right|=-12\,\,\,\,\,\end{array}

Again, we have a result where an absolute value is negative! There is no solution to this equation, or DNE.

In this last video, we show more examples of absolute value equations that have no solutions. https://youtu.be/T-z5cQ58I_g

Solve inequalities containing absolute values

Let’s apply what you know about solving equations that contain absolute values and what you know about inequalities to solve inequalities that contain absolute values. Let’s start with a simple inequality.

∣x∣≤4\left|x\right|\leq 4

This inequality is read, “the absolute value of x is less than or equal to 4.” If you are asked to solve for x, you want to find out what values of x are 4 units or less away from 0 on a number line. You could start by thinking about the number line and what values of x would satisfy this equation. 4 and −4−4 are both four units away from 0, so they are solutions. 3 and −3−3 are also solutions because each of these values is less than 4 units away from 0. So are 1 and −1−1, 0.5 and −0.5−0.5, and so on—there are an infinite number of values for x that will satisfy this inequality. The graph of this inequality will have two closed circles, at 4 and −4−4. The distance between these two values on the number line is colored in blue because all of these values satisfy the inequality. Number line. Closed blue circles on negative 4 and 4. Blue line between closed blue circles. The solution can be written this way: Inequality: −4≤x≤4-4\leq x\leq4 Interval: [−4,4]\left[-4,4\right] The situation is a little different when the inequality sign is “greater than” or “greater than or equal to.” Consider the simple inequality ∣x∣>3\left|x\right|>3. Again, you could think of the number line and what values of x are greater than 3 units away from zero. This time, 3 and −3−3 are not included in the solution, so there are open circles on both of these values. 2 and −2−2 would not be solutions because they are not more than 3 units away from 0. But 5 and −5−5 would work, and so would all of the values extending to the left of −3−3 and to the right of 3. The graph would look like the one below. Number line. Open blue circles on negative three and three. Blue arrow through all numbers less than negative 3. Blue arrow through all numbers greater than 3. The solution to this inequality can be written this way: Inequality: x<−3x<−3 or x>3x>3. Interval: (−∞,−3)∪(3,∞)\left(-\infty, -3\right)\cup\left(3,\infty\right) In the following video, you will see examples of how to solve and express the solution to absolute value inequalities involving both AND and OR. https://youtu.be/0cXxATY2S-k

Writing Solutions to Absolute Value Inequalities

For any positive value of a and x, a single variable, or any algebraic expression:
Absolute Value Inequality Equivalent Inequality Interval Notation
∣x∣≤a\left|{ x }\right|\le{ a} −a≤x≤a{ -a}\le{x}\le{ a} [−a,a]\left[-a, a\right]
∣x∣<a\left| x \right|\lt{a} −a<x<a{ -a}\lt{x}\lt{ a} (−a,a)\left(-a, a\right)
∣x∣≥a\left| x \right|\ge{ a} x≤−a{x}\le\text{−a} or x≥a{x}\ge{ a}  (−∞,−a]∪[a,∞)\left(-\infty,-a\right]\cup\left[a,\infty\right)
∣x∣>a\left| x \right|\gt\text{a} x<−a\displaystyle{x}\lt\text{−a} or x>a{x}\gt{ a}  (−∞,−a)∪(a,∞)\left(-\infty,-a\right)\cup\left(a,\infty\right)
Let’s look at a few more examples of inequalities containing absolute values.

Example

Solve for x. ∣x+3∣>4\left|x+3\right|\gt4

Answer: Since this is a “greater than” inequality, the solution can be rewritten according to the “greater than” rule.

x+3<−4       or       x+3>4 \displaystyle x+3<-4\,\,\,\,\,\,\,\text{or}\,\,\,\,\,\,\,x+3>4

Solve each inequality.

x+3<−4                  x+3>4    −3     −3‾                     −3  −3‾x         <−7                  x         >1x<−7       or      x>1       \begin{array}{r}x+3<-4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x+3>4\\\underline{\,\,\,\,-3\,\,\,\,\,-3}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,-3\,\,-3}\\x\,\,\,\,\,\,\,\,\,<-7\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\,\,\,\,\,\,\,\,\,>1\\\\x<-7\,\,\,\,\,\,\,\text{or}\,\,\,\,\,\,x>1\,\,\,\,\,\,\,\end{array}

Check the solutions in the original equation to be sure they work. Check the end point of the first related equation, −7−7 and the end point of the second related equation, 1.

   ∣x+3∣>4              ∣x+3∣>4∣−7+3∣=4               ∣1+3∣=4       ∣−4∣=4                         ∣4∣=4            4=4                              4=4 \displaystyle \begin{array}{r}\,\,\,\left| x+3 \right|>4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| x+3 \right|>4\\\left| -7+3 \right|=4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| 1+3 \right|=4\\\,\,\,\,\,\,\,\left| -4 \right|=4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| 4 \right|=4\\\,\,\,\,\,\,\,\,\,\,\,\,4=4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4=4\end{array}

Try −10−10, a value less than −7−7, and 5, a value greater than 1, to check the inequality.

     ∣x+3∣>4              ∣x+3∣>4∣−10+3∣>4              ∣5+3∣>4          ∣−7∣>4                        ∣8∣>4              7>4                             8>4 \displaystyle \begin{array}{r}\,\,\,\,\,\left| x+3 \right|>4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| x+3 \right|>4\\\left| -10+3 \right|>4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| 5+3 \right|>4\\\,\,\,\,\,\,\,\,\,\,\left| -7 \right|>4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| 8 \right|>4\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,7>4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,8>4\end{array}

Both solutions check!

Answer

Inequality: x<−7     or     x>1 \displaystyle x<-7\,\,\,\,\,\text{or}\,\,\,\,\,x>1 Interval: (−∞,−7)∪(1,∞)\left(-\infty, -7\right)\cup\left(1,\infty\right) Graph: x 1

Example

Solve for y. 3∣2y+6∣−9<27 \displaystyle \mathsf{3}\left| \mathsf{2}\mathrm{y}\mathsf{+6} \right|-\mathsf{9<27}

Answer: Begin to isolate the absolute value by adding 9 to both sides of the inequality.

3∣2y+6∣−9<27  +9   +9‾3∣2y+6∣        <36 \displaystyle \begin{array}{r}3\left| 2y+6 \right|-9<27\\\underline{\,\,+9\,\,\,+9}\\3\left| 2y+6 \right|\,\,\,\,\,\,\,\,<36\end{array}

Divide both sides by 3 to isolate the absolute value.

3∣2y+6∣‾ <36‾3                  3          ∣2y+6∣<12\begin{array}{r}\underline{3\left| 2y+6 \right|}\,<\underline{36}\\3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,3\,\\\,\,\,\,\,\,\,\,\,\left| 2y+6 \right|<12\end{array}

Write the absolute value inequality using the “less than” rule. Subtract 6 from each part of the inequality.

−12<2y+6<12  −6             −6   −6‾−18 < 2y         <  6 \begin{array}{r}-12<2y+6<12\\\underline{\,\,-6\,\,\,\,\,\,\,\,\,\,\,\,\,-6\,\,\,-6}\\-18\,<\,2y\,\,\,\,\,\,\,\,\,<\,\,6\,\end{array}

Divide by 2 to isolate the variable.

−18‾<2y‾< 6 ‾2           2          2  −9<  y    < 3\begin{array}{r}\underline{-18}<\underline{2y}<\underline{\,6\,}\\2\,\,\,\,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\,\,\,2\,\,\\-9<\,\,y\,\,\,\,<\,3\end{array}

Answer

Inequality: −9<  y  <3 \displaystyle -9<\,\,y\,\,<3 Interval: (−9,3)\left(-9,3\right) Graph: Open dot on negative 9 and open dot on 3, with a line through all numbers between 9 and 3.

In the following video, you will see an example of solving multi-step absolute value inequalities involving an OR situation. https://youtu.be/d-hUviSkmqE In the following video you will see an example of solving multi-step absolute value inequalities involving an AND situation. https://youtu.be/ttUaRf-GzpM In the last video that follows, you will see an example of solving an absolute value inequality where you need to isolate the absolute value first. https://youtu.be/5jRUuiMUxWQ

Identify cases of inequalities containing absolute values that have no solutions

As with equations, there may be instances in which there is no solution to an inequality.

Example

Solve for x. ∣2x+3∣+9≤7\left|2x+3\right|+9\leq 7

Answer: Isolate the absolute value by subtracting 9 from both sides of the inequality.

∣2x+3∣+9 ≤   7                  −9     −9‾       ∣2x+3∣   ≤−2  \displaystyle \begin{array}{r}\left| 2x+3 \right|+9\,\le \,\,\,7\,\,\\\underline{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-9\,\,\,\,\,-9}\\\,\,\,\,\,\,\,\left| 2x+3 \right|\,\,\,\le -2\,\end{array}

The absolute value of a quantity can never be a negative number, so there is no solution to the inequality.

Answer

No solution

Summary

Absolute inequalities can be solved by rewriting them using compound inequalities. The first step to solving absolute inequalities is to isolate the absolute value. The next step is to decide whether you are working with an OR inequality or an AND inequality. If the inequality is greater than a number, we will use OR. If the inequality is less than a number, we will use AND. Remember that if we end up with an absolute value greater than or less than a negative number, there is no solution.

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