CAT Practice : Inequalities

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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?

Quadratic Inequalities

    Q.3: Find the range of x for which (x + 2) (x + 5) > 40.

 

  • Correct Answer
    x < -10 or x > 3

Detailed Solution

There are two ways of trying this one. We can expand and simplify this algebraically.
x2 + 7x + 10 > 40 or x2 + 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 ax2 + 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.

Correct Answer : x < -10 or x > 3

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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.