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Inequalities on GMAT problem solving questions are pretty straightforward. In their most basic form, you solve an inequality almost exactly like you solve an equation, remembering to reverse the inequality sign if you end up multiplying or dividing by a negative number. Nothing earth-shattering there.

But what about more difficult GMAT inequalities involving two variables? Consider an example like this:

Q: If 3 < x < 8 and 5 < y < 11, which of the following represents all of the possible values of xy?
(A) 3 < xy < 11
(B) 8 < xy < 19
(C) 15 < xy , 88
(D) 24 < xy < 55
(E) 33 < xy < 40

Now, there are at least three different ways to get a right answer that are all pretty good, but I’d like to show you a way that I think is the fastest, most efficient, and most versatile way to solve this particular type of GMAT question. The way you do it is to set up an Inequality Table.

The key is to understand that the possible values of xy will be defined by the extreme possible values of x and y individually.

In other words, x can be as small as 3 (well, technically, 3.000000000000….01, but you get the idea) and as large as 8. Likewise, y can be as small as 5 and as large as 11. So, those numbers are the “extremes” that we’ll want to put into columns 1 and 2 of our Inequality Table. In its generic form, then, the inequality table will look like this:

Extreme Values of X Extreme Values of Y Combination of X and Y as Defined by the Problem
3 5 ??
3 11 ??
8 5 ??
8 11 ??

Then, the third column will be whatever combination of x and y the problem is asking you to solve for. In this case, the question wants to know the possible values of x times y, or xy. Therefore, you simply multiply x * y in each of the rows to produce the products in column 3:

x y xy
3 5 15
3 11 33
8 5 40
8 11 88

The answer to the question, then, is simply the extreme values in column 3 — 15 and 88. Therefore, the answer to the question is answer choice C: 15 < xy < 88. Great job if you got that!

The beauty of this mathematical approach is its versatility. Don’t be fooled by the way this example happened to work out — the “extremes” in column 3 won’t always be the values in the first and last row. Consider this variation to the question:

Q: If 3 < x < 8 and 5 < y < 11, which of the following represents all of the possible values of x-y?

You set up the Inequality Table the exact same way, only this time the values you put in column 3 are the differences between x and y in each row:

x y x – y
3 5 -2
3 11 -8
8 5 3
8 11 -3

Notice this time how the extreme values are in rows 2 and 3? The answer to this question, if I had given you answer choices, would therefore be -8 < x-y < 3.

Sure, you can memorize a bunch of inequality rules to try to mentally solve this type of question. Sure, you can make up numbers for the variables and eliminate wrong answer choices until you’re left with the right answer. But at the end of the day, I think you’ll find the use of an Inequality Table faster and more useful on difficult GMAT inequality questions involving two variables. Practice it in your GMAT prep, and go out and Dominate the GMAT!