GMAT Questions | Inequalities

Inequalities is an important topic and one can expect questions in inequalities in the quantitative reasoning section of the GMAT exam. Questions in inequalities appear in both variants - data sufficiency and problem solving. Inequalities, data sufficiency, indices, and absolute value is a potent combination.

A collection of GMAT practice questions in inequalities is given below. Attempt these questions and check whether you have got the correct answer. If you could not solve the question, go to the explanatory answer or the video explanations to learn how to crack the question.

Curated sample GMAT questions from these topics are presented here below. Attempt these questions and check if you have got the correct answer. One can use the explanatory answer or Video explanation made available (wherever provided) to help crack important GMAT quant questions.

1. If "x" is an integer, which of the following inequality(ies) will have a finite range of values of "x" satisfying it(them)?

   1. x² + 5x + 6 > 0
   2. |x + 2| > 4
   3. 9x - 7 < 3x + 14
   4. x² - 4x + 3 < 0
   5. (B) and (D)

   Correct AnswerChoice D
   

   Explanation Video Solution
   Hint to solve this GMAT 650 level Inequalities question

   Step 1: Compute solution set to each of the 4 inequalities given in the answer options.
   Step 2: Two of these inequalities are quadratic expressions, one is computing inequality of absolute values and another one is a linear algebraic expression.
   Step 3: Select those options that have a finite range of values.

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2. What values of 'x' will be the solution to the inequality 15x - \\frac{2}{x}) > 1?

   1. x > 0.4
   2. x < \\frac{1}{3})
   3. - \\frac{1}{3}) < x < 0.4, x > \\frac{15}{2})
   4. -\\frac{1}{3}) < x < 0, x > \\frac{2}{5})
   5. x < -\\frac{1}{3}) or x > \\frac{2}{5})

   Correct AnswerChoice D
   

   Explanation Video Solution
   Hint to solve this GMAT 650 level Inequalities sample question

   Step 1: Rewrite the expression such that the right hand side of the inequality is zero.
   Step 2: Simplify the numerator to get a quadratic expression.
   Step 3: Factorize the numerator and identify zeroes of each of the factors.
   Step 4: Find intervals that will satisfy the condition that the numerator and denominator of the algebraic expression should both be positive or both be negative.
   Step 5: Compute the solution set.

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3. What is the smallest integer that satisfies the inequality \\frac{x - 3}{x^{2} - 8x - 20}) > 0?

   1. -2
   2. 10
   3. 3
   4. -1
   5. 0

   Correct AnswerChoice D
   The smallest integer that satisfies the inequality is -1.

   Explanation Video Solution
   Hint to solve this GMAT 650 level Inequalities practice question

   Step 1: Factorize the quadratic expression in the denominator.
   Step 2: Identify the zeroes of each of the factors of the denominator and the term in the numerator.
   Step 3: Arrange the zeroes in ascending order and identify the intervals to check whether the inequality holds good.
   Step 4: Compute the smallest integers from among the solution set for the inequality.

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4. Is a < b?

   1. a^b < b^a
   2. \\frac{a}{b}) > 1

   Correct AnswerChoice E
   The two statements together are NOT sufficient.

   Explanation Video Solution
   Hint to solve this GMAT 700 level Inequalities DS Question

   Step 1: Make a note of when is the answer to the question 'yes' and when is the answer 'no'.
   Step 2: Evaluate statement 1 alone. Approach - find a counter example. If you can find a counter example, statement 1 is not suficient.
   Step 3: Evaluate statement 2 alone. Approach - find a counter example. If you can find a counter example, statement 1 is not suficient.
   Step 4: If the statements are independently not sufficient, combine the statements and determine sufficiency.

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5. Is | a | > | b |?

   1. \\frac{1}{(a-b)}) > \\frac{1}{(b-a)})
   2. a + b < 0

   Correct AnswerChoice C
   The two statements together are SUFFICIENT to answer the question.

   Explanation Video Solution
   Approach to solve this GMAT 650 level Inequalities DS question

   Step 1: Make a note of when is the answer to the question 'yes' and when is the answer 'no'.
   Step 2: Evaluate statement 1 alone. Understand the implication of statement 1 and check whether we can get a conclusive answer using statement 1. Approach - counter example.
   Step 3: Evaluate statement 2 alone. Approach counter example.
   Step 4: If the statements are independently not sufficient, combine them and determine the answer. For the given statements, should there be a need to combine the statements, counter example will be an ideal approach.

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6. Is a³ > a²?

   1. \\frac{1}{a}) > a
      
   2. a⁵ > a³

   Correct AnswerChoice A Statement 1 alone is sufficient; statement 2 is not.

   Explanation Video Solution
   Approach to solve this GMAT 650 level DS question

   Step 1: Make a note of when is the answer to the question 'yes' and when is the answer 'no'.
   Step 2: Evaluate Statement 1 Alone: Identify the interval in which the statement will hold good if 'a' is positive and evaluate whether we get a conclusive answer to the question. Also, evaluate the range of values in which the statement will hold good if 'a' is negative. Determine whether for negative values of 'a' we get the same definitive answer to the question.
   Step 3: Evaluate Statement 2 Alone: For positive values of 'a' when will a⁵ > a³? In that interval, what is answer to the question. For negative values of 'a' when will a⁵ > a³? For those values of 'a' what is the answer to the question. Do we have a conclusive answer?

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7. Is | a | > a?

   1. a² < a
   2. \\frac{a}{2}) > \\frac{2}{a})

   Correct AnswerChoice A Statement 1 alone is sufficient; statement 2 is not.

   Explanation Video Solution
   Hint to solve this GMAT Inequalities of Absolute Values (Modulus) question

   Step 1: Rephrase the question? For what values of 'a' will | a | > a? Will be easier to find an answer to this question.
   Step 2: Make a note of when is the answer to the question 'yes' and when is the answer 'no'.
   Step 3: Evaluate Statement 1 Alone. It is clear that a² is non-negative. Find the solution set for the inequality, a² < a. Determine whether we get a definite answer to the rephrased question.
   Step 4: Evaluate Statement 2 Alone. Find the solution set for the expression given in statemnet 2 for the possibility that 'a' is positive. Compute the range of values of 'a' that satisfy the inequality in statement 2 when 'a' is negative. Check whether we have a conclusive answer to the question for both positive and negative values of 'a'.

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8. Is 'a' positive?

   1. a - b > 0
   2. 2a - b > 0

   Correct AnswerChoice E
   The two statements together are NOT sufficient.

   Explanation Video Solution
   Hint to solve this GMAT Inequalities sample question

   Step 1: Make a note of when is the answer to the question 'yes' and when is the answer 'no'.
   Step 2: Evaluate Statement 1 Alone. Approach: Counter example.
   Step 3: Evaluate Statement 2 Alone. Approach: Counter example.
   Step 4: Evaluate statements together if required. Approach: Counter example.

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9. Is aⁿ > bⁿ?

   1. a > b
   2. ab < 0

   Correct AnswerChoice E
   The two statements together are NOT sufficient.

   Explanation Video Solution
   Hint to solve this GMAT 650 level Inequalities sample question

   Step 1: Make a note of when is the answer to the question 'yes' and when is the answer 'no'.
   Step 2: Evaluate Statement 1 Alone. Approach: Counter example. Hint - check with positive and negative values for 'b'.
   Step 3: Evaluate Statement 2 Alone. Approach: Counter example. Hint - swap values of 'a' and 'b'.
   Step 4: Evaluate the statements together by reasoning out the two conditions.

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