# Precalculus : Solve Absolute Value Inequalities

## Example Questions

### Example Question #2 : Linear Inequalities

Solve for

Explanation:

In order to solve this equation, we must first isolate the absolute value. In this case, we do it by dividing both sides by  which leaves us with:

When we work with absolute value equations, we're actually solving two equations. So, our next step is to set up these two equations:

and

In both cases we solve for  by adding  to both sides, leaving us with

and

This can be rewritten as

### Example Question #3 : Linear Inequalities

Solve for

Explanation:

When we work with absolute value equations, we're actually solving two equations:

and

Adding  to both sides leaves us with:

and

Dividing by  in order to solve for  allows us to reach our solution:

and

Which can be rewritten as:

### Example Question #1 : Linear Inequalities

Solve for

Explanation:

In order to solve for  we must first isolate the absolute value. In this case, we do it by dividing both sides by 2:

As with every absolute value problem, we set up our two equations:

and

We isolate  by adding  to both sides:

and

Finally, we divide by :

and

### Example Question #5 : Linear Inequalities

Solve for .

Explanation:

Our first step in solving this equation is to isolate the absolute value. We do this by dividing both sides by

.

We then set up our two equations:

and .

Subtracting 4 from both sides leaves us with

and .

Lastly, we multiply both sides by 2, leaving us with :

and .

Which can be rewritten as:

### Example Question #1 : Solve Absolute Value Inequalities

Solve for

Explanation:

We first need to isolate the absolute value, which we can do in two steps:

1. Add 2 to both sides:

2. Divide both sides by 4:

Our next step is to set up our two equations:

and

We can now solve the equations for  by subtracting both sides by 8:

and

and then dividing them by 5:

and

Which can be rewritten as:

### Example Question #1 : Solve Absolute Value Inequalities

Solve the following absolute value inequality:

Explanation:

First we need to get the expression with the absolute value sign by itself on one side of the inequality. We can do this by subtracting seven from both sides.

Next we need to set up two inequalities since the absolute value sign will make both a negative value and a positive value positive.

From here, subtract thirteen from both sides and then divide everything by four.

### Example Question #8 : Linear Inequalities

Solve the following absolute value inequality:

Explanation:

First we need to get the expression with the absolute value sign by itself on one side of the inequality. We can do this by dividing both sides by three.

We now have two equations:

and

So, our solution is

### Example Question #1 : Solve Absolute Value Inequalities

Solve the following inequality:

Explanation:

First we need to get the expression with the absolute value sign by itself on one side of the inequality. We can do this by subtracting two from both sides then dividing everything by three.

Since absolute value signs make both negative and positive values positive we need to set up a double inequality.

Now to solve for  subtract four from each side.

### Example Question #2 : Linear Inequalities

Solve for :

Explanation:

If , then either or based on the meaning of the absolute value function. We have to solve for both cases.

a) subtract 5 from both sides

divide by -2, which will flip the direction of the inequality

Even if we didn't know the rule about flipping the inequality, this answer makes sense - for example, , and .

b) subtract 5 from both sides

divide by -2, once again flipping the direction of the inequality

### Example Question #1 : Solve Absolute Value Inequalities

Solve the absolute value inequality.

Explanation:

First, simplify so that the absolute value function is by itself on one side of the inequality.

.

Note that the symbol flipps when you divide both sides by .

Next, the two inequalities that result after removing the absolute value symbols are

and   .

When you simplify the two inequalities, you get

and .

Thus, the solution is

.