GMAT Inequalities Challenge Problem I.1: Explanation

If you haven’t already looked at the article on the 3 methods you can use to solve GMAT inequalities and Absolute value questions like these, please read through the article. This GMAT Inequalities Challenge problem was posted in the same article. The GMAT Inequalities article covers all the basics you need to understand to solve questions like this. Hence, in this explanation, we assume that you are already familiar with these methods and we will skip over the basics and focus primarily on two methods:

• Algebra
• Distance

The “pick a value” method has been left as an exercise for the readers.

Algebra method

Lets unpack the question’s statement first using Algebra.

Is |x-y| > |x|?

Since both sides are non-negative, let us start by squaring both sides and removing the absolute value.

(x-y)² > x²? => x² + y² – 2xy > x² ? => y² – 2xy > 0 ?

Here, we can try to simplify further, but for now, let us leave it at this. The question essentially becomes

Is |x-y| > |x|? is same as Is y² – 2xy > 0?

Now lets take the statements one by one:

Statement (a)

Given (a) |x+y|<|x-y|

We are given that|x+y|<|x-y| , this tells us a couple of things:

• both LHS and RHS are non-negative, hence as before we can square both sides.
• Both x & y cannot be zero at the same time. Why? If x and y were both zero, we will be “given” 0<0 – which is not possible.

So lets start with squaring the both sides, it is given that:

(x+y)² < (x-y)² => xy < 0

If xy<0, what can we say about the question stem which evaluated to “Is y²-2xy > 0“

Since xy is negative, lets assume xy = -K, where K is positive. So the question becomes

Is y² + 2K >0 ?

Since y² is positive & K is also positive, the answer is a unique YES. So Statement (a) is Sufficient to answer the question.

Statement (a) is SUFFICIENT.

On a cursory look, we may conclude that x and y are opposite signs. That will be a BIG MISTAKE. In statement (b), we are given x = -7y. This means one of two things:

1. Either x and y are of opposite signs OR
2. x and y are both zero [It is critical not to miss this possibility]

1. if x and y are of opposite signs, then xy < 0 and it is similar situation to the Statement (a). And it will answer “YES” to the asked question
2. If x and y are both zero, then the question becomes Is 0 > 0? The answer to this question is a clear “NO”

Hence for Statement (b), there is no unique YES/NO to the asked question. Hence Statement (b) is insufficient. This is a trap in this this GMAT Inequalities Challenge Problem.

Statement (b) is INSUFFICIENT

Hence the answer is A: Only Statement (a) is sufficient to answer the question but not statement (b)

Distance method

From the distance perspective, lets cover a couple of quick basics here:

• |x-y| means distance from x to y (or from y to x)
• |x| means distance from x to 0 (or from 0 to x)
• |x+y| means distance from x to -y (or from -y to x)

Is |x-y| > |x|?

So, the question is asking “is the distance between x and y more than distance between x and 0”.

Is |x-y| > |x|? can be rewritten as: “Is the distance between x & y greater than the distance between x & 0”

Let’s see this from a number line perspective. Consider x to be greater than 0

The yellow zone indicates all the possible values of y where the distance between y and x is more than the distance between x and 0. How? Take any value of y in the yellow zone. On the positive side of the number line, since y is beyond 2x, the |x-y| value will obviously be more than |x|.

On the negative side of the number line, since x has to go beyond 0 to reach to y, it is obvious that |x-y| will be greater than |x|.

A similar diagram can be drawn for x<0 and you will get a similar result. Few things are clear from this diagram:

1. if x and y are opposite signs, |x-y| > |x|
2. If x and y are same sign than |x-y| > |x| WHEN |y| > 2|x|

Now let’s look at the question statements.

Statement (a)

Given (a) |x+y|<|x-y|

Look at the figures below,

From the second diagram we clearly get that the “given” expression of |x-y|>|x+y| is only possible when x and y are opposite signs. So essentially, it is given to us that x and y are opposite signs. From (1) above, if x and y are opposite signs, we get the answer to the question as YES. Hence (a) is SUFFICIENT.

Statement (a) is SUFFICIENT.

On a cursory look, we may conclude that x and y are opposite signs. That will be a BIG MISTAKE.

Statement (b), we are given x = -7y. This means one of two things:

1. Either x and y are of opposite signs OR
2. x and y are both zero [It is critical not to miss this possibility]

1. if x and y are of opposite signs, then xy < 0 and it is similar situation to the Statement (a). And it will answer “YES” to the asked question
2. If x and y are both zero, then the question becomes Is 0 > 0? The answer to this question is a clear “NO”

Hence for Statement (b), there is no unique YES/NO to the asked question. Hence Statement (b) is insufficient.

Statement (b) is INSUFFICIENT

Hence the answer is (A) Only Statement (a) is sufficient to answer the question but not statement (b)

Hopefully, the explanation is detailed enough for you to understand the framework and the solution for this GMAT Inequalities Challenge question. Stay tuned for more questions. Should you have any questions, please feel free to leave a comment or send us an email.