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 x^{2} - |x + 3| + x > 0?

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

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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 x^{2} - x - 3+ x > 0 i.e., x^{2} -3 > 0

x^{2} > 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 x^{2} + x + 3 + x > 0

⇒ x^{2} + 2x + 3 > 0

For any value of x < -3, x^{2} will be positive and geater than 2x even if x is negative.

So, x^{2} + 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 x^{2} - |x + 3| + x, becomes (-3)^{2} - |-3 + 3| + (-3) = 9 - 3 = 6.

Hence at x = -3, x^{2} - |x + 3| + x > will be satisfied

Combining all 3 x ∊ (-3, -√3) ∪ (√3 , ∞) ∪ -3 ∪ (-∞, -3)

which becomes, x ∊ (-∞,-√3) ∪ (√3, ∞)

The question is **"Solve x ^{2} - |x + 3| + x > 0?"**

Choice B is the correct answer.

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