CAT Practice : Inequalities

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This is an excellent question to understand how a polynomial behaves when the variable 'x' takes values between roots. Think about how you can make the polynomial take positive and negative values. What values of 'x' should you substitute in the polynomial?

Inequalities - Integer - Solutions

    Q.13: For how many integer values does the following inequality hold good? (x + 2) (x + 4) (x + 6)........(x + 100) < 0.
    1. 25
    2. 50
    3. 49
    4. 47

 

  • Correct Answer
    Choice A. 25

Explanatory Answer

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Detailed Solution

(x + 2) (x + 4) (x + 6) ........(x + 100) < 0
Now, the above expression will be zero for x = –2, –4, –6, – 8…..–100.
For x > – 2 all the terms will be positive and so, the product will be positive.
For x < – 100, all the terms will be negative and since there are 50 terms (even number), the product will be positive.

Now, if x = – 99, the term x + 100 would be positive, everything else would be negative, so the expression would have 49 negative terms and one positive term. So the product would be negative.

Overall the expression will be negative if there are exactly 49 negative terms, or exactly 47 negative terms, or exactly 45 terms…. Or so on, up to exactly one negative term.

Exactly 49 negative terms =. x = – 99
Exactly 47 negative terms =. x = – 95
Exactly 45 negative terms =. x = – 91
...............
Exactly 1 negative term =. x = – 3

So, x can take values {–3, –7, –11, –15, –19…. –99}. We need to compute how many terms are there in this list.

In other words, how many terms are there in the list {3, 7, 11, ....99}. Now, these terms are separated by 4, so we can write each term as as multiple of 4 + ‘some constant’.

Or 3 = 0 * 4 + 3
7 = 1 * 4 + 3
11 = 2 * 4 + 3
......................
99 = 24 * 4 + 3
We go from 0 * 4 + 3 to 24 * 4 + 3, a total of 25 terms. Choice A.

Correct Answer: 25



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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.