# CAT Practice : Inequalities

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This is the world of inequality (we do not mean inequity). Ignore it at your own peril. You will definitely get questions from this topic, and it forms the basis for much of algebra.
1. ### 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
• Correct AnswerChoice (B).
There are 30 integer values that 'x' can take that satisfy the inequality.
• Integer Solutions
• Medium
2. ### Cubic Inequalities

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

1. (2, ∞)
2. ((1, 2) ∪ (2, ∞)
3. (-∞, 1) ∪ (2, ∞)
4. (-∞, 1)
• Correct AnswerChoice (B).
Within this range, the inequality remains positive, or greater than zero
• Cubic Inequalities
• Hard
3. ### Quadratic Inequalities

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

• Correct Answerx < -10 or x > 3 Correct answer
• Easy
4. ### Integer Roots - Trial and Error

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

• Correct AnswerThere are a total of nine values Correct answer
• Integer Roots
• Medium
5. ### Modulus Inequalities

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

• Correct Answer( -∞, -4) or (2, 4) or ( 10, ∞) Correct answer
• Modulus
• Hard
6. ### Natural Numbers

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

• Correct AnswerThe maximum product is 560. Correct answer
• Natural Numbers
• Easy
7. ### Integers - Polynomials

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

8. ### Modulus - Quadratic

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

• Correct AnswerThe range is (-∞, -5) or (-2, 2) Correct answer
• Modulus
• Hard
9. ### 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?

• Correct Answer(-∞, -2) or (-2, 2) Correct answer
• Hard
10. ### 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?

11. ### Modulus - Tricky Question

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

12. ### 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?

13. ### Inequalities - Integer Solutions

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

14. ### 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?

15. ### Properties of Inequalities

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

1. x $\in (-\infty,-1] \cup [\sqrt{3}, 3)$
2. x $\in (-\infty,-3] \cup [\sqrt{3}, \infty)$
3. x $\in (-4,-3) \cup (4, \infty)$
4. x $\in (-8,-3] \cup [2, \infty)$
• Correct AnswerChoice (B).
• Properties of Inequalities
• Medium
16. ### Properties of Inequalities

Find range of f(x) = x2 – 6x + 14

1. $(-\infty, 8)$
2. $(-\infty, 100)$
3. $(-\infty, 45)$
4. $(5, \infty)$

• Correct AnswerChoice (D).
• Properties of Inequalities
• Medium
17. ### Properties of Inequalities

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

1. $(-\infty,-5) \cup (4, \infty)$
2. $(-\infty,-3] \cup [3, \infty)$
3. $(-\infty,-5] \cup [1, \infty)$
4. $(-\infty,-1] \cup [2, \infty)$
• Correct AnswerChoice (A).
• Properties of Inequalities
• Medium
18. ### Modulus - Tricky

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
19. ### 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
20. ### 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
21. ### 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. ${\frac{119}{4}}$
22. ### 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
23. ### Inequalities - Range

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

1. [0,${\infty}$ )
2. [-4,${\infty}$ )
3. All real values of x
4. [4,${\infty}$ )
24. ### Inequalities - Max and Min Values

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

1. 3 and 1
2. 3 and 0
3. 4 and 0
4. 4 and 1
• Correct AnswerChoice (A).
• Inequalities - Max and Min Values
• Medium

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