# Algebra 1 : How to solve absolute value equations

## Example Questions

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### Example Question #1 : How To Solve Absolute Value Equations

Solve the absolute value equation:

(no solution)

Explanation:

An equation that equates two absolute value functions allows us to choose one of the absolute value functions and treat it as the constant. We then separate the equation into the "positive" version,  , and the "negative" version,. Solving each equation, we obtain the solutions,  and , respectively.

### Example Question #2 : How To Solve Absolute Value Equations

Solve for x.

x = 4, 7

x = 1, 7

No solution

x = –7, 1

x = –3, 4

x = –7, 1

Explanation:

First, split  into two possible scenarios according to the absolute value.

Looking at , we can solve for x by subtracting 3 from both sides, so that we get x = 1.

Looking at , we can solve for x by subtracting 3 from both sides, so that we get x = –7.

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.

1, 5

2, 4

0

2, 5

1, 4

1, 5

Explanation:

|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:

or

or

or

Explanation:

Because of the absolute value signs,

or

Subtract 2 from both sides of both equations:

or

or

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

Solve for :

Explanation:

There are two answers to this problem:

and

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

If , evaluate .

Explanation:

An absolute value expression differs from a normal expression only in its sign. Instead of being a positive or negative quantity, an absolute value represents a scalar distance from zero, so it does not have a sign. For example,  is the same as  because both represent a value 2 units away from zero. In this problem,  equals , or 5.  equals 8. The final answer is  or 40.

Explanation:

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

What is the solution set of this equation?

Explanation:

To find a solution, subtract  first to isolate the absolute value expression.

There is no value of  that makes this true, as no number has a negative absolute value. The equation has no solution.

Explanation:

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

Solve for

or

or

or

or

Explanation:

can be rewritten as the compound statement:

or

Solve each separately to obtain the solution set:

So either  or

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