# GMAT Math : Solving inequalities

## Example Questions

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### Example Question #31 : Inequalities

Give the solution set of the inequality

Explanation:

To solve a quadratic inequality, move all expressions to the left first:

The boundary points of the solution set will be the points at which:

: that is, , or

: that is, .

None of these values will be included in the solution set, since equality is not allowed by the inequality symbol.

Test the intervals

by choosing a value in each interval and testing the truth of the inequality.

: Test

True; include the interval

: Test

False; exclude the interval .

: Test

True; include the interval .

The solution set is .

### Example Question #32 : Inequalities

Give the solution set of the inequality

Explanation:

To solve a rational inequality, move all expressions to the left first:

The boundary points of the solution set will be the points at which:

- that is, ;

; and

- that is, .

None of these values will be included in the solution set, since equality is not allowed by the inequality symbol.

Test the intervals

by choosing a value in each interval and testing the truth of the inequality.

- test

False - exclude

- test

True - include

- test

False - exclude

- test

True - include

The solution set is .

### Example Question #33 : Inequalities

Give the solution set of the inequality

Explanation:

The square of a real number must be nonnegative, so this is a true statement regardless of the value of . The solution set is the set of all real numbers

To solve a quadratic inequality, move all expressions to the left first

The boundary points of the solution set will be the points at which:

; that is, ; or,

; that is,

These values will be included in the solution set, since equality is allowed by the inequality symbol.

Test the intervals

by choosing a value in each interval and testing the truth of the inequality.

: test

False - exclude this interval

: test

True - include this interval

: test

False - exclude this interval

is the solution set.

### Example Question #34 : Inequalities

Solve for :

Explanation:

can be rewritten as the inequality

(note the change in direction of the inequality symbols)

This is the set .

### Example Question #35 : Inequalities

Solve the following inequality: