GMAT Math : Solving inequalities

Study concepts, example questions & explanations for GMAT Math

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Example Questions

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Example Question #31 : Inequalities

Give the solution set of the inequality

Possible Answers:

Correct answer:

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 

Possible Answers:

Correct answer:

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 

Possible Answers:

Correct answer:

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 :

Possible Answers:

Correct answer:

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:

 

Possible Answers:

Correct answer:

Explanation:

Like any other equation, we solve the inequality by first grouping like terms. Grouping the  terms on the left side of the equation and the constants on the right side of the equation, we have:

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