Inequalities play a crucial role in Algebra for CAT 2026, especially in higher-difficulty CAT questions. Practice exam-oriented CAT algebra important questions covering linear, quadratic, and modulus inequalities. These are built around repeated CAT concepts and logical reasoning patterns. For complete preparation, also practice Quadratic Equations, Functions, and Logarithms, as they often integrate with inequalities in CAT.
Q3. Let f and g be a real valued functions defined as f(x) = x2 + 8 and g(x) = f(x – 2) + f(x + 2) – 36. For how many integral values of ‘x’, g(x) < 0?
Q5. A farmer wants to fence his rectangular field to protect his crops from animals. The cost of fencing the field is Rs. 300 per meter on one side, and Rs. 150 per meter along three other sides. If the area of the rectangular field is 1350 sq. m, then what is the lowest possible cost (in Rs.) of fencing the field?
Q6. If y = log(7 – a) (2x2 + 2x + a + 3) for all real values of x, then which of the following is NOT a possible integral value of ‘a’?
Q9. If P3 – 8P2 + P + 42 < 0, then what is the smallest positive integer value of P?
Q10. A, B, C and D are four positive numbers which satisfy the equation A + 2B + 3C + 4D = 36. Find the maximum possible value of AB2CD2.
Q11. Find the number of integral solutions for the inequality (|x – 1| –5)(|x + 2| –4) < 0.
Q12. For non-negative real numbers x and y, 3x + 4y ≤ 16 and 5x + 2y ≤ 10. What is the maximum possible value of (4x + 10y)?
Q13. How many integral values of ‘x’, which is a real number, satisfy the equations x2 – 4x – 96 < 0 and x2 > 4?
Q14. Find the number of distinct pairs of non-negative integers (x, y) satisfying |3 – 2xy| < |3x – 2y| < 9.
Q15. Let M and N be number of integral solutions to ||x| – 2022| < 10 and |2030 – |y|| < 12. What is the value of |M – N|?
Q18. If a and b are non-negative integers such that a + 8 = c, b + 2 = c and a + b > c + 3, then the minimum possible value of (2a + 3b – c) equals:
Q19. If p > 3 and q < ‒1, then which of the following statements is always true?
Q22. The number of distinct pairs of integers (m, n) satisfying |2 – mn| < m – n ≤ 2 is:
Q26. If ‘n’ is an integer such that (log4 n – 3) (log3 n – 4) < 0, then the number of possible values of ‘n’ is:
Q28. (x – 2) (x – 4) (x – 6) (x – 8) …………….. (x – 20) ≤ 0. How many integer values of x satisfy the given equation?
Q29. What is the number of integer values that x can take for both the inequations x2 – 6x – 72 < 0 and x2 – 20x +100 > 0?
Q30. If w + x + y + z = 1, where w, x, y, z are non-negative, then the maximum value of wxyz/(1 – w) (1 – x)(1 – y)(1 – z) is:
Q32. Number of integer values for which the inequality (3x – 1) (x – 3) > (2x2 – 8x + 2) does not hold true is/are:
Q33. The number of integral values of x satisfying the inequality |6x – 9| < 15 – 4x is:
Q34. If |x2 – 11x + 30| > x2 – 11x + 30, then which of the following statements is true?
Q36. Let f(x) = (x – 2)(x – 1)(x + 3)(x + 5), then which of the following is true regarding the number of integral roots of f(x) > 0 and f(x) < 0 respectively?
Q38. If ||x – 5| – 3| ≥ 1, find the number of integral values of ‘x’ which doesn’t satisfy the inequality.
Q39. What is the number of integer values that x can take for both the equations x2 – 6x – 72 < 0 and x2 – 20x + 100 > 0?
Q40. If |x – 3| < 2 and |x| > 2, for how many integers will |x – 1| < 3?
Q41. What is the highest value of ‘x’ for which x3 – 14x2 + 49x – 36 < 0?
Q44. If A, B, C and D are distinct real numbers in geometric progression such that for every real X,
(X2 CB)+(AC)+(BD)=AD, then X must satisfy which of the following conditions?
Q47. If (x+1)(y-1)=16, x > 0, y > 1, then:
Q49. lf 2a + 3b + 6c = 66, where a, b and c are positive real numbers, then find the maximum value of a6 b2 c3.
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