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CAT Quantitative Aptitude | CAT Algebra: Inequalities Questions

A CAT Algebra question from Inequalities that appears in the Quantitative Aptitude section of the CAT Exam consists of concepts: Range of Inequalities, Modulus functions, Possible solutions and so on. The topic also involves linear and quadratic equations, finding roots, polynomials, functions and more. Inequalities is a crucial topic for CAT. Having a good foundation in this subject can help a student tackle questions in Coordinate Geometry, Functions. In CAT Exam, one can generally expect to get 2~3 questions from Inequalities. Make use of 2IIMs Free CAT Questions, provided with detailed solutions and Video explanations to obtain a wonderful CAT score. If you would like to take these questions as a Quiz, head on here to take these questions in a test format, absolutely free.

  1. CAT Inequalities - Integer Solutions

    How many positive integer values can x take that satisfy the inequality (x - 8) (x - 10) (x - 12).......(x - 100) < 0?

    1. 25
    2. 30
    3. 35
    4. 40
    Choice B
    30

  2. CAT Inequalities - Cubic Inequalities

    Solve the inequality: x3 – 5x2 + 8x – 4 > 0?

    1. (2, ∞)
    2. (1, 2) ∪ (2, ∞)
    3. (-∞, 1) ∪ (2, ∞)
    4. (-∞, 1)
    Choice B
    (1, 2) ∪ (2, ∞)

  3. CAT Inequalities - Quadratic Inequalities

    Find the range of x for which (x + 2) (x + 5) > 40?

    x < -10 or x > 3

  4. CAT Inequalities - Integer Roots, Trial and Error

    How many integer values of x satisfy the inequality x( x + 2)(x + 4)(x + 6) < 200?

    There are a total of nine values

  5. CAT Inequalities - Modulus Inequalities

    Find the range of x where ||x - 3| - 4| > 3?

    ( -∞, -4) or (2, 4) or ( 10, ∞)

  6. CAT Inequalities - Natural Numbers

    The sum of three distinct natural numbers is 25. What is the maximum value of their product?

    The maximum product is 560.

  7. CAT Inequalities - Integers, Polynomials

    If x (x + 3) (x + 5) (x + 8) < 250, how many integer values can x take?

    11 values

  8. CAT Inequalities - Modulus, Quadratic

    (|x| - 2) (x + 5) < 0. What is the range of values x can take?

    The range is (-∞, -5) or (-2, 2)

  9. CAT Inequalities and Quadratic Inequalities

    a and b are roots of the equation x2 - px + 12 = 0. If the difference between the roots is at least 12, what is the range of values p can take?

    (-∞, -2) or (-2, 2)

  10. CAT Inequalities - Integer Solutions

    If a, b, c are distinct positive integers, what is the highest value a × b × c can take if a + b + c = 31?

    1. 1080
    2. 1200
    3. 1024
    4. 1056
    Choice A
    1080

  11. CAT Inequalities - Modulus

    a, b, c are distinct natural numbers less than 25. What is the maximum possible value of |a – b| + |b – c| – |c – a|?

    1. 44
    2. 46
    3. 23
    4. 21
    Choice A
    44

  12. CAT Inequalities - Maximum Possible Value

    Consider integers p, q such that – 3 < p < 4, – 8 < q < 7, what is the maximum possible value of p2 + pq + q2?

    1. 60
    2. 67
    3. 93
    4. 84
    Choice B
    67

  13. CAT Inequalities - Integer Solutions

    For how many integer values does the following inequality hold good? (x + 2) (x + 4) (x + 6)........(x + 100) < 0?

    1. 25
    2. 50
    3. 49
    4. 47
    Choice A
    25

  14. CAT Inequalities - Maximum possible value

    If a, b, c are integers such that – 50 < a, b, c < 50 and a + b + c = 30, what is the maximum possible value of abc?

    1. 1000
    2. 5000
    3. 4410
    4. 4560
    Choice C
    4410

  15. CAT Inequalities - Properties of Inequalities

    Solve x2 - |x + 3| + x > 0?

    1. x ∊ (-∞,-1] ∪ [√3, 3)
    2. x ∊ (-∞,-√3) ∪ (√3, ∞)
    3. x ∊ (-4,-3) ∪ (4, ∞)
    4. x ∊ (-8,-3] ∪ [2, ∞)
    Choice B
    x ∊ (-∞,-√3) ∪ (√3, ∞)

  16. CAT Inequalities - Properties of Inequalities

    Find range of f(x) = x2 – 6x + 14?

    1. (-∞, 8)
    2. (-∞, 100)
    3. (-∞, 45)
    4. (5, ∞)
    Choice D
    (5, ∞)

  17. CAT Inequalities - Properties of Inequalities

    Solve :\\frac{(x – 4) (x+3)}{(x + 4) ( x +5)}\\) > 0?

    1. x ∊ (-∞,-5) ∪ (-4 , -3) ∪ (4, ∞)
    2. x ∊ (-∞,-5) ∪ (4, ∞)
    3. x ∊ (-4,-3) ∪ (4, ∞)
    4. x ∊ (-5,-3] ∪ [4, ∞)
    Choice A
    x ∊ (-∞,-5) ∪ (-4 , -3) ∪ (4, ∞)

  18. CAT Inequalities and Modulus

    Consider three distinct positive integers a, b, c all less than 100. If |a - b| + |b - c| = |c – a|, what is the maximum value possible for b?

    1. 98
    2. 99
    3. 50
    4. 100
    Choice A
    98

  19. CAT Inequalities - Integers

    Consider integers m, n such that -5 < m < 4 and -3 < n < 6. What is the maximum possible value of m2 - mn + n2?

    1. 65
    2. 60
    3. 50
    4. 61
    Choice D
    61

  20. CAT Inequalities - Integers

    Consider integers p, q, r such that |p| < |q| < |r| < 40. P + q + r = 20. What is the maximum possible value of pqr?

    1. 3600
    2. 3610
    3. 3510
    4. 3500
    Choice C
    3510

  21. CAT Inequalities - Minimum Value

    What is the minimum value of f(x) = x2 – 5x + 41?

    1. \\frac{139}{4}\\)
    2. \\frac{149}{4}\\)
    3. \\frac{129}{4}\\)
    4. \\frac{119}{4}\\)
    Choice A
    \\frac{139}{4}\\)

  22. CAT Inequalities - Roots

    x4 – 4x3 + ax2 – bx = 1 = 0 has positive real roots. What is the maximum possible value of a + b?

    1. 20
    2. 12
    3. 8
    4. 10
    Choice D
    10

  23. CAT Inequalities - Range

    |x3 – 3x + 5| > -4. What range of x satisfies this?

    1. [0,∞)
    2. [-4, ∞)
    3. All real values of x
    4. [4,∞)
    Choice C
    All real values of x

  24. CAT Inequalities - Max and Min Values

    What are the maximum and minimum possible values for \\frac{|x+y|}{|x|+|y|}\\) + \\frac{|z+y|}{|z|+|y|}\\) + \\frac{|z+x|}{|x|+|z|}\\)?

    1. 3 and 1
    2. 3 and 0
    3. 4 and 0
    4. 4 and 1
    Choice A
    3 and 1

The Questions that follow, are from actual CAT papers. If you wish to take them separately or plan to solve actual CAT papers at a later point in time, It would be a good idea to stop here.


  1. CAT 2020 Question Paper Slot 2 - Inequalities

    If x and y are non-negative integers such that x + 9 = z, y + 1 = z and x + y < z + 5, then the maximum possible value of 2x + y equals


  2. CAT 2018 Question Paper Slot 2 - Number Theory | Inequalities

    The smallest integer n for which 4n > 1719 holds, is closest to

    1. 33
    2. 39
    3. 37
    4. 35
    Choice B
    39

  3. CAT 2018 Question Paper Slot 2 - Polynomials | Inequalities

    The smallest integer n such that n3 - 11n2 + 32n - 28 > 0 is (TITA)


  4. CAT 2018 Question Paper Slot 2 - Inequalities

    If a and b are integers such that 2x2 - ax + 2 > 0 and x2 - bx + 8 ≥ 0 for all real numbers x, then the largest possible value of 2a - 6b is (TITA)


  5. CAT 2017 Question Paper Slot 1 - Inequalities

    For how many integers n, will the inequality (n – 5) (n – 10) – 3(n – 2) ≤ 0 be satisfied? (TITA)


The Questions that follow, are from actual XAT papers. If you wish to take them separately or plan to solve actual XAT papers at a later point in time, It would be a good idea to stop here.


  1. XAT 2018 Question Paper - QADI

    If 2 ≤ |x – 1|×|y + 3| ≤ 5 and both x and y are negative integers, find the number of possible combinations of x and y.

    1. 4
    2. 5
    3. 6
    4. 8
    5. 10
    Choice E
    10

The Questions that follow, are from actual IPMAT papers. If you wish to take them separately or plan to solve actual IPMAT papers at a later point in time, It would be a good idea to stop here.


  1. IPMAT 2020 Question Paper - IPM Indore Quants

    Consider the following statements:
    (i) When 0 < x < 1, then \\frac{1}{1+x}) < 1 - x + x2
    (ii) When 0 < x < 1, then \\frac{1}{1+x}) > 1 - x + x2
    (iii) When -1 < x < 0, then \\frac{1}{1+x}) < 1 - x + x2
    (iv) When -1 < x < 0, then \\frac{1}{1+x}) > 1 - x + x2
    Then the correct statements are

    1. (i) and (ii)
    2. (ii) and (iv)
    3. (i) and (iv)
    4. (ii) and (iii)
    Choice C
    (i) and (iv)

  2. IPMAT 2019 Question Paper - IPM Indore Quants

    If x ∈ (a, b) satisfies the inequality \\frac{x - 3}{x^{2} + 3x + 2} \geq 1,\\) then the largest possible value of b - a is

    1. 3
    2. 1
    3. 2
    4. No real values of x satisfies the inequality
    Choice B
    1


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