# GRE Subject Test: Math : Absolute Value Inequalities

## Example Questions

### Example Question #1 : Absolute Value Inequalities

or

Explanation:

The first thing we must do is get the absolute value alone:

When we're working with absolute values, we are actually solving two equations:

and

Fortunately, these can be written as one equation:

If you feel more comfortable solving the equations separately then go ahead and do so.

To get  alone, we added  on both sides of the inequality sign

### Example Question #1 : Absolute Value Inequalities

There is no solution.

There is no solution.

Explanation:

Because Absolute Value must be a non-negative number, there is no solution to this Absolute Value inequality.

### Example Question #1 : Absolute Value Inequalities

The weight of the bowling balls manufactured at the factory must be  lbs. with a tolerance of  lbs.  Which of the following absolute value inequalities can be used to assess which bowling balls are tolerable?

Explanation:

The following absolute value inequality can be used to assess the bowling balls that are tolerable:

### Example Question #4 : Absolute Value Inequalities

and

and

There is no solution.

and

and

Explanation:

and

and

and

and

and

Explanation:

### Example Question #1 : Absolute Value Inequalities

and

and

and

There is no solution.

and

Explanation:

and

and

and

and

and

Explanation:

### Example Question #1 : Absolute Value Inequalities

A type of cell phone must be less than 9 ounces with a tolerance of 0.4 ounces. Which of the following inequalities can be used to assess which cell phones are tolerable? (w refers to the weight).

Explanation:

The Absolute Value Inequality that can assess which cell phones are tolerable is:

### Example Question #1 : Absolute Value Inequalities

Solve for x:

Explanation:

Step 1: Separate the equation into two equations:

First Equation:
Second Equation:

Step 2: Solve the first equation

Step 3: Solve the second equation

The solution is

### Example Question #1 : Absolute Value Inequalities

Which of the following expresses the entire solution set of ?

and

and

Explanation:

Before expanding the quantity within absolute value brackets, it is best to simplify the "actual values" in the problem. Thus  becomes:

From there, note that the absolute value means that one of two things is true:  or . You can therefore solve for each possibility to get all possible solutions. Beginning with the first:

means that:

For the second:

means that:

Note that the two solutions can be connected by putting the inequality signs in the same order: