# 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.10: If a, b, c are distinct positive integers, what is the highest value a × b × c can take if a + b + c = 31?
1. 1080
2. 1200
3. 1024
4. 1056

Choice A. Maximum product is 1080

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## Detailed Solution

The sum of three numbers is given; the product will be maximum if the numbers are equal.

${{a + b + c} \over 3} > \root 3 \of {abc}$
So, if a + b + c is defined, abc will be maximum when all three terms are equal. In this instance, however, with a, b, c being distinct integers, they cannot all be equal.

So, we need to look at a, b, c to be as close to each other as possible.
a = 10, b =10, c = 11 is one possibility, but a, b, c have to be distinct. So, this can be ruled out.
The close options are,
a, b, c: 9, 10, 12; product = 1080
a, b, c: 8, 11, 12: product = 1056
Maximum product = 1080

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## More questions from Inequalities

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.