# 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 - Modulus - Tricky

Q.11: a, b, c are distinct natural numbers less than 25. What is the maximum possible value of |a – b| + |b – c| – |c – a|?
1. 44
2. 46
3. 23
4. 21

Choice A. 44

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## Detailed Solution

For any two points M, N on the number line representing numbers m, n the distance MN = | m - n|.
So, for three points, P, Q and R on the number line |p – q|, |q – r|, |r – p| are distances between three pairs of points on the number line.

In this case, we are trying to find maximum value of |a – b| + |b – c| – |c – a|. If b lies between a and c, the above value would be zero. So, b should not be between a and c.
The best case scenario would be if a, c were very close to each other and far from b. Let us try b = 24, a = 1, c = 2.
In this case |a – b| + |b – c| – |c – a| = 23 + 22 – 1 = 44. This is the maximum possible value.

We could also have b = 1, a = 24, c = 23,
|a – b| + |b – c| – |c – a| = 23 + 22 – 1 = 44.

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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.