GMAT Tip: Inequalities…and then some… Part One and Two

student taking the gmat

Part One

When it comes to tricky quantitative questions, inequalities take the prize for serving as one of the biggest deceivingly easy questions. Test takers should just treat inequalities as algebraic “equal to” equations with just a < or > symbol in place of the = sign, right?

If only it was so simple. Inequalities are designed to assess your critical thinking skills, working beyond solving for a value of a variable to considering the range of variable possibilities. Furthermore, inequalities questions are set up as data sufficiency questions the majority of the time. Smart test takers know to be on their toes.

Let’s start with a 500 level quantitative inequality question:

Is n between 0 and 1?

Statement 1: n^2 < n
Statement 2: n^3 > 0

Start by evaluating the question. What does it mean when n is between 0 and 1? It means n is not an integer. It is also a positive fraction or decimal, such as ½ or 0.25.

Now that we know what we are working with, so let’s move on to Statement 1. What would make n^2 less than n?

The variable n cannot possibly be a negative number, because by squaring it n becomes a positive number.

The only way the n^2 could be less than n is if n was a fraction, like ½. When ½ is squared (½ * ½) it becomes ¼, which is smaller than ½.

Indeed, Statement 1 is sufficient to prove that n lies between 0 and 1.

Okay, so next up is Statement 2, where n^3 > 0. If we have the same mindset as Statement 1, we work to assess the options.

If n^3 is greater than 0, then it cannot possibility be negative. But, then, does n definitely lie between 0 and 1?

Not necessarily. The expression n^3 means that, sure, if n = ½ then (½)^3 is ⅛ and therefore greater than 0. The variable n could also be 1, or 2, which would make n^3 greater than 0, but not a fraction or decimal.

Statement 2 is not sufficient. The correct answer to this question is (A).

When looking at inequality questions, the key to getting the right answer is being flexible in your thinking.

But inequalities are rarely this simple. Often, these trickster questions are complicated by absolute value, multiple variables, and square roots.

Part Two

While not as simple as they seem, inequality questions can be huge time wasters that keep scores back at the 600 level.

As with any GMAT quantitative concept, there is always the possibility that a curve ball will be thrown your way. Thought that just assessing > or < was the gist of all inequality questions. Think again.

One of the big “add-ins” that we see with inequality questions is absolute value. Remember that, by definition, absolute value is the magnitude of a number without its sign. So, when starting to work with inequality questions with absolute value notation, we need to recognize that there are likely two (possibly more) values we need to consider.

Let’s start by looking at an example problem that incorporates both inequalities, multiple variables, and absolute value:

If y is an integer and y = |x| + x, is y = 0? 

Statement 1: x < 0 

Statement 2: y < 1 

Before jumping right into the decision tree in evaluating Statement 1 and Statement 2, we need to first take a hard look at what the question is asking us the test taker to determine.

For y = |x| + x, when we should understand that y can never be negative. For the second part of the question, it is helpful to figure out that y = 0 when x is a negative number. Take a parallel example:

If x = -2, when we plug it into the problem we find that y = |-2| + (-2) -> 2 + (2) = 0. Therefore, whenever x = a negative number, y will be equal to zero… no matter if it is -(½), -2, or -1966.

By getting a handle on the question on its own, it is so much easier for us to try to determine whether or not Statement 1 and/or Statement 2 is sufficient or not sufficient. For this particular problem, it is immediately clear that when x < 0 – Statement 1 – is sufficient.

Moving on to Statement 2, another curve ball is thrown. Wait, now we have to consider the y value? Hold fast to your first assessment. If y is never negative, as we figured in our beginning analysis of the question, and that it is also an integer, then the only possibility with Statement 2 is that y = 0, same as with Statement 1. Statement 2 is also sufficient, indicating that the right answer choice is (D).

By getting our bearings with the question, first, before looking at the Statements and corresponding values, we are less likely to be confused and make the gross mistake of using a loose “plug-in numbers” method. Making this same smart strategic move on inequality questions with absolute values with set you up for success to do well on these slightly trickier quantitative questions.

Stay tuned for Part 3 when we take a look at inequalities with square roots.

This article originally appeared in

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