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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.

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
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, ∞)
CAT Inequalities - Quadratic Inequalities

Find the range of x for which (x + 2) (x + 5) > 40?
x < -10 or x > 3
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
CAT Inequalities - Modulus Inequalities

Find the range of x where ||x - 3| - 4| > 3?
( -∞, -4) or (2, 4) or ( 10, ∞)
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.
CAT Inequalities - Integers, Polynomials

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

11 values
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)
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)
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
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
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
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
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
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, ∞)
CAT Inequalities - Properties of Inequalities

Find range of f(x) = x² – 6x + 14?
1. (-∞, 8)
2. (-∞, 100)
3. (-∞, 45)
4. (5, ∞)
Choice D
(5, ∞)
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, ∞)
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
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
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
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}\\)
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
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
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.

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
23
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
CAT 2018 Question Paper Slot 2 - Polynomials | Inequalities
The smallest integer n such that n³ - 11n² + 32n - 28 ＞ 0 is (TITA)
8
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)
36
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)
11

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.

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.

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)
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