CAT Quantitative Aptitude Questions | CAT Algebra - Inequalities questions

CAT Questions | Algebra | Inequalities - Cubic Inequalities

The question is about finding the solution of a cubic inequality. Our task is to find the solution of the cubic inequality. Inequalities are crucial to understand many topics that are tested in the CAT exam. Having a good foundation in this subject can help us tackle questions in Coordinate Geometry, Functions, and most importantly in Algebra. A range of CAT questions can be asked based on this simple concept.

Question 2: Solve the inequality x3 – 5x2 + 8x – 4 > 0.

  1. (2, ∞)
  2. (1, 2) ∪ (2, ∞)
  3. (-∞, 1) ∪ (2, ∞)
  4. (-∞, 1)

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Explanatory Answer

Method of solving this CAT Question from Algebra - Inequalities : We all know (or should soon) how to solve a quadratic inequality. How about a cubic? Try to see if you can factorise the equation, and see how the equation behaves around the roots.

Let a, b, c be the roots of this cubic equation
a + b + c = 5
ab + bc + ca = 8
abc = 4
This happens when a = 1, b = 2 and c = 2 {This is another approach to solving cubic equations}.

The other approach is to use polynomial remainder theorem
If you notice, sum of the coefficients = 0
=> P(1) = 0
=> (x - 1) is a factor of the equation. Once we find one factor, we can find the other two by dividing the polynomial by (x - 1) and then factorizing the resulting quadratic equation.
(x - 1) (x - 2) (x - 2) > 0

Let us call the product (x - 1) (x - 2) (x - 2) as a black box.
If x is less than 1, the black box is a –ve number
If x is between 1 and 2, the black box is a +ve number
If x is greater than 2, the black box is a +ve number

Since we are searching for the regions where the black box is a +ve number, the solution is as follows: 1 < x < 2 OR x > 2

The question is "Solve the inequality x3 – 5x2 + 8x – 4 > 0."

Hence the answer is "(1, 2) ∪ (2, ∞)"

Choice B is the correct answer.


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