# GMAT Math : Solving inequalities

## Example Questions

### Example Question #21 : Inequalities

Solve for :

Explanation:

The first step to solving this inequality is to subtract 18 from all three segments of the equation:

8 (- 18)  < -5x + 18 ( - 18) < 103 ( - 18)

-10 < -5x < 85

We then divide the entire equation by -5. According to inequality rules, when we multiply or divide an inequality by a negative number, we must flip the inequality signs.

(-10/-5) > (-5x/-5) > (85/-5)

2 > x > -17 or in conventional layout -17 < x < 2

### Example Question #22 : Inequalities

Solve:

or

or

Explanation:

In order to solve this inequality, we must split it into two inequalities:

and

We then solve each, remembering of course that, when multiplying or dividing by a negative number, we must switch the direction of the inequality sign.

and

Therefore:

### Example Question #23 : 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, ;

;

; 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 .

: test

False; exclude the interval .

The solution set of the inequality is .

### Example Question #24 : Inequalities

Give the solution set of the inequality

Explanation:

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

Since the square of any real number is nonnegative, there is no value of  for which this is true. The solution set is the empty set.

### Example Question #25 : Inequalities

Give the solution set of the inequality

Explanation:

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

The only boundary point for the intervals in the solution set is the point at which  - that is, . This will be excluded from the solution set, as the symbol does not allow equality. But for any other value of , the square of the number must be positive. Therefore, the solution set is

.

### Example Question #26 : Inequalities

Give the solution set of the inequality

Explanation:

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

Since the square of any number must be nonnegative, it follows that for any

and the solution set is the set of all real numbers, .

### Example Question #27 : Inequalities

True or false:

Statement 1:

Statement 2:

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

First, solve the inequality:

The boundary points of the intervals to be tested are the points at which:

; that is, , or

; that is, .

Therefore, the intervals  should be tested; do so by testing one value in each interval in the inequality and testing its truth:

- test

False - reject this interval.

- test

True - accept this interval.

- test

False - reject this interval.

The solution set is the interval ; therefore,  if and only if .

From Statement 1 alone, we know that ; this provides insufficient information, since, for example,  and  both fall in this range, but only the former is a solution of . For similar reasons, Statement 2 alone provides infufficient information.

Assume both statements are true. Together, the two statements are euivalent to saying that . Therefore, it holds that , or , and it can be established that .

### Example Question #28 : Inequalities

Find the solution set of the inequality

Explanation:

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

The square of a real number cannot be less than 0, so

, the only solution.

### Example Question #29 : Inequalities

Find the solution set of the inequality

Explanation:

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

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

### Example Question #30 : Inequalities

Solve for :

Explanation:

If , then one of two things is true.

Either

(note the change in direction of the inequality symbols)

or

(note the change in direction of the inequality symbols)

The set is .

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