The title of this post is: Flipping the sign to inequalities. The only difference to simplifying inequalities compared to normal equations is the times when we have to flip the sign. There are two consistent times when we must always flip the sign:

1. When we multiple by a negative number.
2. When we divide by a negative number.

While these situations may seem straight forward, the GMAT has found fun ways to increase the difficulty. Let’s look at an advanced Data Sufficiency question:

1. Is ?

1)
2)

As we evaluate the first statement, we see that the base of the exponents is the same . Since this inequality is just like an equation, we can drop like bases. However, do you know what happens when you square a fraction? If you square a number between 0 and 1, the number actually gets smaller. Thus, as we look at Statement 1, if we drop the fractional base, we have to flip the sign as well!

(To help clarify: . If we calculate the equation, we see the answer is – which is mathematically correct. If we drop the base without flipping the sign, the inequality reads , which isn’t mathematically accurate.)

Thus, Statement 1 simplifies to: . As we further simplify the inequality, we move our numbers around and achieve or or . Since , it is also greater than -4: Sufficient.

Let’s look at statement 2: Quadratics and inequalities are difficult when combined – the squared variable results in two possible solutions. For this example, a savvy test takers notices we can factor out from the equation. Translating this to . As we look at this situation, we know that either is negative (less than zero) or is negative (less than zero) – but not both. Here we probably have to test both situations.

Test 1: and – one positive and one negative (check out the blog post regarding binomials for further reading on the sign issue). If we solve for y in the above situation, we get the outcome of and . Is this possible? is both greater than 0 and less then -3? Nope. Thus, this scenario is not the right answer.

Test 2: and . If we solve for in this situation, we get the outcome of and . Is this possible? Yes! We can re-write this as . Or, in other words, Statement 2 is sufficient by itself!

The right answer to the above Data Sufficiency question? Either statement on its own is sufficient!

Let’s quickly review the times we need to flip the inequality:

1. When we multiple by a negative number.
2. When we divide by a negative number.
3. When we remove exponential bases that are between 0 and 1
4. When we are dealing with a quadratic equation (there will be two solutions: and ) – don’t forget that binomials usually have two solutions!

Good luck as you practice with inequalities. This was a complicated post! Don’t worry if it doesn’t set in immediately. Print out this screen and read it a couple times – you’ll get it. This is just one of the advanced math topics included among the Kaplan GMAT math material in the newly revised course. If you’ve gotten beyond the intermediate questions and algebra topics, you may be ready to work through advanced topics like this in preparation for the toughest questions on test day.

Brian Fruchey
Kaplan GMAT

What is the value of ?
(1)
(2)

We have two variables, and once we get both statements, we’ll have two equations, so we’ll be able to solve for . The answer is (C), or the third Data Sufficiency answer choice—together the statements are sufficient. If you’ve figured this out, that’s awesome. You’ve discovered how to save a lot of time on Test Day.

But I always tell students not to get trigger-happy. Before you pick (C), keep in mind that the GMAT often gives you situations in which we can get sufficiency with just one equation, or when two won’t be enough. Here are three of those situations:

The Vanishing Variable

What is the value of ?
(1)
(2)

Both equations have two variables, so how could one possibly be sufficient to solve for ? Let’s play with Statement (1) a bit so we can isolate . Distribute the right side of the equation to get . Then we can subtract from both sides, and poof! We have a single variable equation. We certain can solve for . The answer is (A), statement 1 alone is sufficient to answer the question. So before you settle for (C), ask yourself if you can eliminate a variable from one equation.

Solving for a Relationship

What is the value of ?
(1)
(2)

When the GMAT asks you to solve for a relationship between variables (a sum, difference, product, or quotient), ask yourself, Can I manipulate one of the statements to solve for that relationship? If you can do this, you’ll only need one equation for sufficiency. In this case, no amount of manipulating of Statement 1 can do the trick, but let’s play with Statement 2. Divide both sides by 3, and you get . We still don’t know what or is, but we do know what is. The answer is (B), statement 2 alone is sufficient.

The Disguised Twin

What is the value of ?
(1)
(2)

It seems like we have everything we need to pick (C) here. Two equations, two variables, we’re golden. Except dig a little deeper; Statement (2) should cause Déjà vu. Add 10y to both sides of the second equation and divide everything by 2, and you’ll discover that the two equations are identical—just dressed up a little differently. Since we really have just one equation with two variables, we have a recipe for insufficiency. The answer is (E), there is not enough information within these statements to answer the question, no matter how you use them or combine them.

Going forward

So you don’t necessarily have to solve these systems of equations when you see them in a DS question, but you will have to do some detective work before you pick (C). As we advise students in our newly revised GMAT courses, ask yourself the following questions when assessing this type of problem: Can you make a variable vanish? Can you solve for the desired relationship by manipulating an equation? Are these equations really different? When you know the anatomy of the test, you score higher.

Ben Leff
Kaplan GMAT

To gain this grasp, we must start with the most basic idea of what exactly data sufficiency is asking you to do. A data sufficiency question is not centered around the actual solution. Rather, data sufficiency tests whether you are able to reach a single answer to the question.

You will be given a question and two statements. You must figure out if the information in the statements, alone or together, is sufficient to answer the question being asked. In other words, based on the statements could you solve the problem if you wanted?

The five answer choices never change and all refer to which statement(s) are sufficient: Statement One but not Statement Two; Statement Two but not Statement One; the Statements together are sufficient but not alone; each Statement is sufficient on its own; or the Statements are not sufficient together nor alone.

This means that you can avoid doing quite a bit of work when encountering a data sufficiency problem. For example, once you have set up an equation, you have no reason to actually solve it as long as you can see that you could solve it enough to have a clear answer to whatever you are asked.

So, next time you see a data sufficiency problem, do not let it intimidate you. Rather, tell yourself that you are lucky to be seeing such a problem. Unlike the problem solving portion of the test, you may be able to avoid doing as much actual math. This, in turn, saves you time for problems that genuinely do take longer to complete and leads directly to a higher score.

Bret Ruber
Kaplan GMAT

But that students frustration soon eases when I explain the key to completing GMAT math problems within the two minute time-frame: choose an approach. Not use an approach. Choose an approach.

Every GMAT problem can be handled in at least two, but usually more, ways. All of the approaches will eventually lead to the correct answer, but only one of them will get the test taker there in less than 120 seconds. And the real secret of the GMAT, is that it is not only trying to test math ability. Sure, math knowledge is a prerequisite to doing well on test day, but that alone is not enough. GMAT problems are designed to reward the student that considers all of the possible ways to approach a problem and then selects the one that is the most effective at reaching the correct answer in less than two minutes.

So, next time you do a GMAT practice problem, do not just start solving using the first method of which you think. Take a few seconds, consider your options and be proactive — choose the best approach in order to maximize your time (and ultimately your score).

Bret Ruber
Kaplan GMAT

Mistake #1: Combining statements when unnecessary

This is done when a test-taker looks at both statements and says “Yes, if I have both pieces of information, then I can figure out the answer, so together the statements are sufficient.” However, you must remember that you’re also asked if either statement ALONE is enough to answer the question. Understanding the differences among all five answer choices in itself can be a boost to your quantitative score. As you look at each individual statement, ask yourself, “is this enough?” Once you can definitively answer yes or no, you are then closer to an answer to the data sufficiency problem.

Mistake #2: Over-calculating

Since you may not need to calculate an actual value for a data sufficiency question, you should avoid going into the calculation step unless absolutely necessary. For example, if dealing with a statement like:

Instead of trying to plow through with the calculations as you might have to do in a problem-solving question, recognize that you have one variable in this equation (x), and that this is solvable. So if this shows up in a data sufficiency question, the answer to the question “is this enough?” is yes, and again you are closer to solving your data sufficiency question.

Though data sufficiency questions look very abstract, there’s a hidden beauty involved in solving them. Practice these while taking on the mindset of “is this enough?” to maximize your time-management ability for the GMAT.

Arthur Ahn
Kaplan GMAT

In case you haven’t committed them to memory yet, here are those familiar choices:

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
(C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question asked.
(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Let’s say you run into the following problem:

1. Is positive?
(1)
(2)

If you, like most students, look at statement (1) first, you’ll probably say to yourself, “Well, if x is greater than 5, then x must be positive!” and you’d be right: statement (1) is definitely sufficient to answer the question. But before we move on to statement (2), what does this mean for our (A)-(E) answer choices? Well, since statement (1) is sufficient, we can eliminate all choices that would require statement (1) to be insufficient, and that’s choices (B), (C), and (E).

Similarly, if you had a question where statement (1) was NOT sufficient by itself, you could immediately axe choices (A) and (D), since both of those choices require statement (1) to be “sufficient ALONE” to answer the question. And what if NEITHER statement ALONE is sufficient? Well then, we’d be down to (C) and (E) as our only possibilities, with a final answer hinging on the sufficiency of the statements in combination. (The system even works if you look at statement (2) before statement (1), you just have to eliminate slightly different choices. Try it!)

Memorizing this simple method is a cornerstone to mastering Data Sufficiency; using it means you never have to waste valuable time deciphering the intricacies of the question type itself, freeing up valuable time and effort for mathematics and critical thinking. Plus, as with any elimination strategy, it makes guessing much more efficient. If you can eliminate 2-3 answer choices and end up with a 33-50% chance to guess correctly on Data Sufficiency questions—rather than a 20% chance with a blind (A)-(E) guess—it really adds up in your final score.

Adam Grey
Kaplan GMAT

You Got to Know When to Hold Them…

As you are working through some of the most difficult Data Sufficiency problems, we always stress not to do any more calculations then you need to. However, some questions require a hefty amount of manual calculations. If you know you will be able to get the right answer but it might take you a bit longer than you like – that is ok! But, realize you’ll have to go faster on another question. While some questions require more work than others, with enough practice, you will know which questions you should spend your time.

…You Got to Know When to Walk Away…

We’ll all been there. We work through a question and we get stuck at some point. If you find that you’re stuck on a data sufficiency question, realize that you can stare at the question for another 30 seconds or quickly evaluate the parts of the question you do know – i.e. if you know for sure Statement 1 or 2 is sufficient or insufficient. From what you know, a quick guess is far superior to spending another couple minutes hoping an epiphany will come.

…And Know When to Run…

On the extremely rare occasion (hopefully!), there may be a question that you’re just not sure how to approach. If you don’t understand the question (or if you understand the question but are not sure where to go with it), you should guess and move on, quickly. It is better to guess at 1 minute instead of 2 minutes – and, even better to guess at 30 seconds instead of 1 minute.

…You Never Count Your Money Sitting At The Table…

This may be the most important statement – don’t look behind you as you are working through the test. Perhaps you guessed on a question or two. Perhaps you just remembered how to answer a previous question. Perhaps you thought the last question was entirely too easy and that might mean you aren’t doing as well as you typically do. Perhaps you won’t do well enough on this test and be relegated to complete and utter obscurity?!?! Stop! You must always focus on the hand (or question) in front of you. Nothing good happens when you think about previous questions. Stay focused on the task at hand.

…There’ll Be Time Enough for Counting, When the Dealing (or the test) is Done.

Once the test is over, then you can count the questions that could have been. However, surprisingly, you won’t actually care anymore. You’ll get your fantastic score and start dreaming about the acceptance letters that you’re going to be getting… Best of luck as you continue studying!

Brian Fruchey
Kaplan GMAT