The document discusses an inequality involving the product of terms of the form (x + a number) where the numbers range from 2 to 100. It explains that for the product to be negative, there needs to be an odd number of negative terms. This will occur when x is equal to -99, -97, -95, and so on, down to -3. There are 25 such values that x can take. Therefore, the number of integer values for which the inequality holds is 25.

1. Inequalities

2. Inequalities
For how many integer values does the following inequality hold
good? (x + 2) (x + 4) (x + 6)........(x + 100) < 0.
(a) 25 (b) 50
(c) 49 (d) 47

3. Inequalities
(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.
For how many integer values does the following inequality hold
good? (x + 2) (x + 4) (x + 6)........(x + 100) < 0.

4. Inequalities
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.
For how many integer values does the following inequality hold
good? (x + 2) (x + 4) (x + 6)........(x + 100) < 0.

5. Inequalities
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.
For how many integer values does the following inequality hold
good? (x + 2) (x + 4) (x + 6)........(x + 100) < 0.

6. Inequalities
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)
For how many integer values does the following inequality hold
good? (x + 2) (x + 4) (x + 6)........(x + 100) < 0.

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