# 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: $$|a|=-a$$

## 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 $$|x|-6=4$$.

Solution:

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 $$|x-3|=8$$.

Solution:

$$x-3=8$$ 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 $$x=-5$$.

Now, check the solutions:

$$x=11$$ → $$|11-3|=|8|=8$$

$$x=-5$$ → $$|-5-3|=|-8|=8$$

The answers are $$11$$ and $$-8$$.

## Exercises for Absolute Value Equations

### Solve each absolute value equation.

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

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