CAT Quantitative Aptitude Questions | CAT Algebra - Inequalities questions

CAT Questions | Algebra | Properties of Inequalities

The question is about solving an inequality. We need to solve an inequality which have both modulus and power two term. 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 15: Solve x2 - |x + 3| + x > 0?

  1. x ∊ (-∞,-1] ∪ [√3, 3)
  2. x ∊ (-∞,-√3) ∪ (√3, ∞)
  3. x ∊ (-4,-3) ∪ (4, ∞)
  4. x ∊ (-8,-3] ∪ [2, ∞)

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Explanatory Answer

Method of solving this CAT Question from Algebra - Inequalities : This is an excellent inequality question comprising of both quadratic and modulus!!

Anything in side the modulus can be +ve, -ve or 0.

Case i) x + 3 is positive
If x + 3 > 0 ⇒ x > -3
Then equation is in the form x2 - x - 3+ x > 0 i.e., x2 -3 > 0
x2 > 3 ⇒ x < -√3 and x > √3

But x > -3, thus the range is (-3, -√3) ∪ (√3 , ∞)

Case ii) x + 3 is negative
If x + 3 < 0 ⇒ x < -3
Then equation is in the form x2 + x + 3 + x > 0
⇒ x2 + 2x + 3 > 0

For any value of x < -3, x2 will be positive and geater than 2x even if x is negative.
So, x2 + 2x + 3 > 0, will be satisfied for all the values of x < -3,
The range is (-∞, -3)

Case iii) x + 3 is 0
⇒ x = -3
Then the equation x2 - |x + 3| + x, becomes (-3)2 - |-3 + 3| + (-3) = 9 - 3 = 6.

Hence at x = -3, x2 - |x + 3| + x > will be satisfied
Combining all 3 x ∊ (-3, -√3) ∪ (√3 , ∞) ∪ -3 ∪ (-∞, -3)
which becomes, x ∊ (-∞,-√3) ∪ (√3, ∞)

The question is "Solve x2 - |x + 3| + x > 0?"

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

Choice B is the correct answer.


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