# CAT Practice : Inequalities

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The Question might seem very simple, but there is a catch!!

## Inequalities - Max and Min Values

Q.24:What are the maximum and minimum possible values for ${\frac{|x+y|}{|x|+|y|} + \frac{|z+y|}{|z|+|y|}\frac{|z+y|}{|z|+|y|}}$
1. 3 and 1
2. 3 and 0
3. 4 and 0
4. 4 and 1

Choice A. 1

## Detailed Solution

We know that |x + y| < |x| + |y|.

So, each of these fractions lies between 0 and 1. So, all three added together should go from 0 to 3?

Is that the case? Is 2IIM running questions that are this simple?

When would this be 3? If x, y and z all have the same sign. Each fraction would be 1 and we would get 3 overall. Spot on! So, the maximum value is 3.

When can this go to zero.

When x = -y this fraction goes to 0. When x and y have opposite signs, the first term would go to zero. Likewise for the second and third terms as well.

So, what is the catch.

Among x, y, and z at least two will have the same sign.

So, of the three terms maximum of two can go to zero. One will be +1.

So, the minimum total overall = 1 (not 3)

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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.