CAT Practice : Inequalities

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. Correct answer
• Explanatory Answer
• Integer Solutions
• Medium

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 Correct answer
• Explanatory Answer
• Cubic Inequalities
• Hard

Quadratic Inequalities

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

• Correct Answerx < -10 or x > 3 Correct answer
• Explanatory Answer
• Quadratic Inequalities
• Easy

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
• Explanatory Answer
• Integer Roots
• Medium

Modulus Inequalities

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

• Correct Answer( -∞, -4) or (2, 4) or ( 10, ∞) Correct answer
• Explanatory Answer
• Modulus
• Hard

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
• Explanatory Answer
• Natural Numbers
• Easy

Integers - Polynomials

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

• Correct Answer11 values Correct answer
• Explanatory Answer
• Polynomials
• Medium

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
• Explanatory Answer
• Modulus
• Hard

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
• Explanatory Answer
• Quadratic Inequalities
• Hard

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?

• Correct Answer1080 Correct answer
• Explanatory Answer
• Integer Solutions
• Medium

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|?

• Correct Answer44 Correct answer
• Explanatory Answer
• Modulus
• Hard

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?

• Correct Answer67 Correct answer
• Explanatory Answer
• Maximum
• Medium

Inequalities - Integer Solutions

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

• Correct Answer25 Correct answer
• Explanatory Answer
• Integer Solutions
• Hard

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?

• Correct Answer4410 Correct answer
• Explanatory Answer
• Maximum
• Hard

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).
  Correct answer
• Explanatory Answer
• Properties of Inequalities
• Medium

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).
  Correct answer
• Explanatory Answer
• Properties of Inequalities
• Medium

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).
  Correct answer
• Explanatory Answer
• Properties of Inequalities
• Medium

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

• Correct AnswerChoice (A).
  Correct answer
• Explanatory Answer
• Modulus - Tricky
• Medium

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

• Correct AnswerChoice (D).
  Correct answer
• Explanatory Answer
• Inequalities - Integers
• Medium

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

• Correct AnswerChoice (C).
  Correct answer
• Explanatory Answer
• Inequalities - Integers
• Medium

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. ${\frac{119}{4}}$

• Correct AnswerChoice (A).
  Correct answer
• Explanatory Answer
• Inequalities - Range
• Medium

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

• Correct AnswerChoice (D).
  Correct answer
• Explanatory Answer
• Inequalities - Roots
• Medium

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,${\infty}$ )

• Correct AnswerChoice (C).
  Correct answer
• Explanatory Answer
• Inequalities - Range
• Medium

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).
  Correct answer
• Explanatory Answer
• Inequalities - Max and Min Values
• Medium