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?
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
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?"
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