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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 following questions are from IPMAT Rohtak and Indore sample papers. If you want to take these questions as a mock please click below.

IPMAT Rohtak Sample Paper Mock
IPMAT Indore Sample Paper Mock

Please note that the explanation button will take you to the IPMAT solution page.

IPMAT 2020 Sample Paper - IPM Rohtak Quants,Inequalities

The set of all real numbers x for which x² - |x + 2 |+ x > 0, is
1. (-∞, -2) ∪ (2, ∞)
2. (-∞, -√2) ∪ (√2, ∞)
3. (-∞, -1) ∪ (1, ∞)
4. (√2, ∞)
Choice B
(-∞, -√2) ∪ (√2, ∞)
IPMAT 2019 Question Paper - IPM Indore Quants,Inequalities

For all real values of x, \\frac{3x^{2} - 6x + 12}{x^{2} + 2x + 4}\\) lies between 1 and k, and does not take any value above k. Then k equals

1
IPMAT 2019 Question Paper - IPM Indore Quants,Inequalities

If \|x| < 100\\) and \|y| < 100\\), then the number of integer solutions of (x, y) satisfying the equation 4x + 7y = 3 is

29
IPMAT 2019 Question Paper - IPM Indore Quants,Inequalities

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
IPMAT 2019 Question Paper - IPM Indore Quants,Inequalities

If a, b, c are real numbers a² + b² + c² = 1, then the set of values ab + bc + ca can take is:
1. [-1,2]
2. [-\\frac{1}{2}\\), 2]
3. [-1,1]
4. [-\\frac{1}{2}\\), 1]
Choice D
[-\\frac{1}{2}\\), 1]
IPMAT 2019 Question Paper - IPM Indore Quants,Inequalities

The inequality \\log _{2} \frac{3x - 1}{2 - x} < 1\\) holds true for
1. x ∈ (\\frac{1}{3}\\), 1)
2. x ∈ (\\frac{1}{3}\\), 2)
3. x ∈ (0, \\frac{1}{3}\\)) ∪ (1,2)
4. x ∈ (-∞, 1)
Choice A
x ∈ (\\frac{1}{3}\\), 1)
IPMAT 2019 Question Paper - IPM Indore Quants,Inequalities

The set of values of x which satisfy the inequality 0.7^{2x² - 3x + 4} < 0.343 is
1. (\\frac{1}{2}\\), 1)
2. (\\frac{1}{2}\\), ∞)
3. (-∞, \\frac{1}{2}\\))
4. (-∞, \\frac{1}{2}\\)) ∪ (1, ∞)
Choice D
(-∞, \\frac{1}{2}\\)) ∪ (1, ∞)
IPMAT 2019 Question Paper - IPM Indore Quants,Inequalities

Determine the greatest number among the following four numbers
1. 2³⁰⁰
2. 3²⁰⁰
3. 2¹⁰⁰ + 3¹⁰⁰
4. 4¹⁰⁰
Choice B
3²⁰⁰

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 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
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