The question is about Integer solutions. Our task is to identify the integers that satisifies 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 13: For how many integer values does the following inequality hold good? (x + 2) (x + 4) (x + 6)........(x + 100) < 0?

- 25
- 50
- 49
- 47

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(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.

The question is **"For how many integer values does the given inequality hold good? (x + 2) (x + 4) (x + 6)........(x + 100) < 0?"**

Choice A is the correct answer.

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