CAT Quantitative Aptitude Questions | CAT Algebra - Inequalities questions

CAT Questions | Algebra | Properties of Inequalities

The question is for finding the solution of a inequality. We need to solve the inequality which is in the form of fraction. 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 17: Solve :\\frac{(x – 4) (x+3)}{(x + 4) ( x +5)}\\) > 0?

  1. x ∊ (-∞,-5) ∪ (-4 , -3) ∪ (4, ∞)
  2. x ∊ (-∞,-5) ∪ (4, ∞)
  3. x ∊ (-4,-3) ∪ (4, ∞)
  4. x ∊ (-5,-3] ∪ [4, ∞)

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Explanatory Answer

Method of solving this CAT Question from Algebra - Inequalities : Solve for x in the traditonal approach!!

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

Consider this as \\frac{a * b}{c * d}\\) > 0
Clearly, here a < b < c < d
Therefore for \\frac{a * b}{c * d}\\) to be positive (that is, > 0),
all of a,b,c,d must be -ve or
all of a,b,c,d must be positive or
any two of a,b,c,d must be negative and other two positive. Since a < b < c < d, a and b must be negative

So, we have 3 cases for \\frac{(x – 4) (x+3)}{(x + 4) ( x + 5)}\\) > 0

Case i) (x – 4) , (x+3) , (x + 4) , ( x + 5) must be positive.
To satisfy this the smallest number needs to be positive.
(x – 4) > 0 ; x >4

Case ii) (x – 4) , (x+3) , (x + 4) , ( x + 5) must be negative.
To satisfy this the largest number needs to be negative.
(x + 5) < 0 ; x <-5

Case iii) (x – 4) , (x+3) must be negative and (x + 4) , ( x + 5) must be positive.
(x – 4) , (x+3) must be negative
To satisfy this the largest number needs to be negative.
(x + 3) < 0 ; x <-3
(x + 4) , ( x + 5) must be positive.
To satisfy this the smallest number needs to be positive.
(x + 4) > 0 ; x >-4
Combining these two, -4 < x < -3

Combining all x < -5, -4 < x < -3, x > 4,
Hence range is (-∞,-5) ∪ (-4 , -3) ∪ (4, ∞)

The question is "Solve :\\frac{(x – 4) (x + 3)}{(x + 4) ( x + 5)}\\) > 0?"

Hence the answer is "x ∊ (-∞,-5) ∪ (-4 , -3) ∪ (4, ∞)"

Choice A is the correct answer.


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