CAT Practice : Inequalities

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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.

Cubic Inequalities

    Q.2: Solve the inequality x3 – 5x2 + 8x – 4 > 0.
    A. (2, ∞)
    B. (1, 2) ∪ (2, ∞)
    C. (-∞, 1) ∪ (2, ∞)
    D. (-∞, 1)

 

  • Correct Answer
    Choice B. (1, 2) ∪ (2, ∞). Within this range, the inequality remains positive, or greater than zero.

Detailed Solution

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

Answer Choice (B) is the correct answer.

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More questions from Inequalities

  1. Inequalities - Integer Solutions
  2. Cubic Inequalities
  3. Quadratic Inequalities
  4. Integer Roots - Trial and Error
  5. Modulus Inequalities
  6. Natural Numbers
  7. Integers - Polynomials
  8. Modulus - Quadratic
  9. Quadratic Inequalities
  10. Inequalities - Integer Solutions
  11. Modulus - Tricky Question
  12. Maximum Possible Value
  13. Inequalities - Integer solutions
  14. Maximum Possible Value
Inequalities are crucial to understand many topics that are tested in the CAT. Having a good foundation in this subject will make us tackling questions in Coordinate Geometry, Functions, and most importantly in Algebra much more comfortable.