The question is about a quadratic inequality. Our task is to find the range of the given 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 3: Find the range of x for which (x + 2) (x + 5) > 40?

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### Explanatory Answer

##### Method of solving this CAT Question from Algebra - Inequalities : We know how to solve the inequality if its greater or lesser than zero. But what happens when its greater or lesser than some non zero value? Can you reduce it to a familiar form?

There are two ways of trying this one. We can expand and simplify this algebraically.

x^{2} + 7x + 10 > 40 or x^{2} + 7x - 30 > 0

(x + 10) (x - 3) > 0

The roots are -10 and +3.

=> x should lie outside the roots.

Now, what is this based on?

There is a simple thumb rule for solving quadratic inequality

For any quadratic inequality ax^{2} + bx + c < 0

Factorize it as a(x - p) ( x - q) < 0

Whenever a is greater than 0, the above inequality will hold good if x lies between p and q.

a(x - p) (x - q) will be greater than 0, whenever x does not lie between p and q. In other words x should lie in the range ( -∞, p) or (q, ∞).

Now, coming back to the question (x + 10) (x - 3) > 0 Or, x < -10 or x > 3

**Second method:**

5 * 8 = 40, -8 * -5 = 40

So, if x + 2 > 5 this will hold good => x > 3

If x + 2 is less than -8 also, this will hold good => x < -10.

The first method is far more robust.

The question is **"Find the range of x for which (x + 2) (x + 5) > 40?"**

##### Hence the answer is "x < -10 or x > 3"