## 17Calculus Precalculus - Solving Absolute Value Equations

##### 17Calculus

Solving Absolute Value Equations

On this page, we cover how to solve equations (with an equal sign) that contain absolute values. There is one key idea that you need to keep in mind as you solve these equations. To get started, let's watch this video.

### MIP4U - Absolute Value Equations [9min-30secs]

video by MIP4U

It is important to notice one important detail that he did in this video. The key idea is that, if you have only one absolute value expression in the equation, move all terms that do not contain absolute value to one side of the equal sign, leaving only the absolute value term on the other side. Only after doing that, can you set up the two equations without absolute values and solve.

Now you should be ready for some practice problems. After that, you will be ready to solve absolute value inequalities on the next page.

Practice Solving Equations

Basic

Unless otherwise instructed, find all values of $$x$$ that solve each equation.

$$\abs{x-2}=4$$

Problem Statement

Find the values of $$x$$ that solve $$|x-2|=4$$

$$x=-2, ~ x=6$$

Problem Statement

Find the values of $$x$$ that solve $$|x-2|=4$$

Solution

### Dr Chris Tisdell - 2087 video solution

video by Dr Chris Tisdell

$$x=-2, ~ x=6$$

Log in to rate this practice problem and to see it's current rating.

$$|3x-2|=4$$

Problem Statement

Find the values of $$x$$ that solve $$|3x-2|=4$$.

$$x=-2/3, ~ x=2$$

Problem Statement

Find the values of $$x$$ that solve $$|3x-2|=4$$.

Solution

### Dr Chris Tisdell - 2088 video solution

video by Dr Chris Tisdell

$$x=-2/3, ~ x=2$$

Log in to rate this practice problem and to see it's current rating.

$$|3x+2| = 8$$

Problem Statement

Find the values of $$x$$ that solve the equation $$|3x+2| = 8$$.

Solution

### PatrickJMT - 2093 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

$$|7-2x| = 10$$

Problem Statement

Find the values of $$x$$ that solve the equation $$|7-2x| = 10$$.

$$x=-3/2, x=17/2$$

Problem Statement

Find the values of $$x$$ that solve the equation $$|7-2x| = 10$$.

Solution

### PatrickJMT - 2094 video solution

video by PatrickJMT

$$x=-3/2, x=17/2$$

Log in to rate this practice problem and to see it's current rating.

$$|3x-2| + 4 = 4$$

Problem Statement

Find the values of $$x$$ that solve the equation $$|3x-2| + 4 = 4$$.

Solution

### Freshmen Math Doctor - 2560 video solution

Log in to rate this practice problem and to see it's current rating.

$$|x+1| + 6 = 2$$

Problem Statement

Find the values of $$x$$ that solve the equation $$|x+1| + 6 = 2$$.

Solution

### Freshmen Math Doctor - 2561 video solution

Log in to rate this practice problem and to see it's current rating.

$$3|2x-1| = 21$$

Problem Statement

Find the values of $$x$$ that solve the equation $$3|2x-1| = 21$$.

Solution

### 2562 video solution

Log in to rate this practice problem and to see it's current rating.

$$|3x| = 18$$

Problem Statement

Find the values of $$x$$ that solve $$|3x| = 18$$.

$$x = 6$$ and $$x = -6$$

Problem Statement

Find the values of $$x$$ that solve $$|3x| = 18$$.

Solution

### 2886 video solution

$$x = 6$$ and $$x = -6$$

Log in to rate this practice problem and to see it's current rating.

$$|5x + 7| = 42$$

Problem Statement

Find the values of $$x$$ that solve $$|5x + 7| = 42$$.

$$x = 7$$ and $$x = -49/5$$

Problem Statement

Find the values of $$x$$ that solve $$|5x + 7| = 42$$.

Solution

### 2887 video solution

$$x = 7$$ and $$x = -49/5$$

Log in to rate this practice problem and to see it's current rating.

$$|2x - 3 | = 9$$

Problem Statement

Find the values of $$x$$ that solve $$|2x - 3 | = 9$$.

$$x = 6$$ and $$x = -3$$

Problem Statement

Find the values of $$x$$ that solve $$|2x - 3 | = 9$$.

Solution

### 2888 video solution

$$x = 6$$ and $$x = -3$$

Log in to rate this practice problem and to see it's current rating.

$$\displaystyle{ \left| \frac{4x+6}{5} \right| = 6}$$

Problem Statement

Find the values of $$x$$ that solve $$\displaystyle{ \left| \frac{4x+6}{5} \right| = 6}$$.

$$x = 6$$ and $$x = -9$$

Problem Statement

Find the values of $$x$$ that solve $$\displaystyle{ \left| \frac{4x+6}{5} \right| = 6}$$.

Solution

### 2889 video solution

$$x = 6$$ and $$x = -9$$

Log in to rate this practice problem and to see it's current rating.

$$2|3x - 4| + 5 = 27$$

Problem Statement

Find the values of $$x$$ that solve $$2|3x - 4| + 5 = 27$$.

$$x = 5$$ and $$x = -7/3$$

Problem Statement

Find the values of $$x$$ that solve $$2|3x - 4| + 5 = 27$$.

Solution

### 2890 video solution

$$x = 5$$ and $$x = -7/3$$

Log in to rate this practice problem and to see it's current rating.

$$3|4x - 1| - 5 = 16$$

Problem Statement

Find the values of $$x$$ that solve $$3|4x - 1| - 5 = 16$$.

$$x = 2$$ and $$x = -3/2$$

Problem Statement

Find the values of $$x$$ that solve $$3|4x - 1| - 5 = 16$$.

Solution

### 2891 video solution

$$x = 2$$ and $$x = -3/2$$

Log in to rate this practice problem and to see it's current rating.

$$|x-4| + 8 = 3$$

Problem Statement

Find the values of $$x$$ that solve $$|x-4| + 8 = 3$$

no solution

Problem Statement

Find the values of $$x$$ that solve $$|x-4| + 8 = 3$$

Solution

### 2892 video solution

no solution

Log in to rate this practice problem and to see it's current rating.

$$|7x + 2| = 4x + 11$$

Problem Statement

Find the values of $$x$$ that solve $$|7x + 2| = 4x + 11$$.

$$x = 3$$ and $$x = -13/11$$

Problem Statement

Find the values of $$x$$ that solve $$|7x + 2| = 4x + 11$$.

Solution

### 2893 video solution

$$x = 3$$ and $$x = -13/11$$

Log in to rate this practice problem and to see it's current rating.

$$|x + 3| = |x - 11|$$

Problem Statement

Find the values of $$x$$ that solve $$|x + 3| = |x - 11|$$.

Hint

This seems more difficult than it really is. Try all four possible cases with positive and negative signs, then solve each one. After that, look back at your work and notice that really only two cases are required.

Problem Statement

Find the values of $$x$$ that solve $$|x + 3| = |x - 11|$$.

$$x = 4$$

Problem Statement

Find the values of $$x$$ that solve $$|x + 3| = |x - 11|$$.

Hint

This seems more difficult than it really is. Try all four possible cases with positive and negative signs, then solve each one. After that, look back at your work and notice that really only two cases are required.

Solution

The four cases are
1. Both sides positive
2. Both sides negative
3. Left side positive and right side negative
4. Right side positive and left side negative.
Cases 1 and 2 are the same. Cases 3 and 4 are the same.

### 2894 video solution

$$x = 4$$

Log in to rate this practice problem and to see it's current rating.

Intermediate

$$|2x-1| = 3|4-8x| - 12$$

Problem Statement

Find the values of $$x$$ that solve $$|2x-1| = 3|4-8x| - 12$$ giving your answer in exact form.

Solution

### Michael Penn - 3884 video solution

video by Michael Penn

Log in to rate this practice problem and to see it's current rating.

$$|3x+6| + |x-1| = 10$$

Problem Statement

Find the values of $$x$$ that solve $$|3x+6| + |x-1| = 10$$ giving your answer in exact form.

Solution

### Michael Penn - 3885 video solution

video by Michael Penn

Log in to rate this practice problem and to see it's current rating.

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.

Links and banners on this page are affiliate links. We carefully choose only the affiliates that we think will help you learn. Clicking on them and making purchases help you support 17Calculus at no extra charge to you. However, only you can decide what will actually help you learn. So think carefully about what you need and purchase only what you think will help you.

We use cookies on this site to enhance your learning experience.