Most of the Quantitative questions you will see on the GMAT involve arithmetic to some extent. The GMAT frequently increases the difficulty level of questions by combining various topics, such as absolute value, inequalities, exponents, and fractions, and often incorporating these topics into questions that involve algebra, geometry, or proportions. You will be in a much stronger position on Test Day if you have these definitions and operations down cold, freeing up your brain to focus on the more complex critical thinking tasks these difficult and often abstract questions require. This chapter will guide you through the basics and give you practice applying arithmetic concepts to GMAT questions ranging from the simple to the most advanced.
As is often the case with GMAT Quantitative questions, a couple of different yet related concepts are involved within this single question. Here, the question gives a range of possible values of x. So you’re looking at inequalities, dealing with both positive and negative numbers. The question itself asks about absolute value, a different topic, but very much related to inequalities, as you’ll see later in this chapter.
Each answer choice contains one variable, x, which makes the question seem a little more straightforward. However, note that a few of the answer choices contain x2, which introduces more complexity, in that you must understand the behavior of exponential expressions.
Absolute value is the measure of a number’s distance from zero on the number line. (Because 5 and −5 are the same distance from zero—5 units—both numbers have the same absolute value: 5.) When squaring a term, the result will always be non-negative. So even though the possible values of x are both positive and negative, certain answer choices will test your ability to recognize this property of terms raised to even exponents.
Here, the test maker rewards those who understand how paying Attention to the Right Detail is important to answering the question correctly. Despite the basic nature of some of the math topics, the GMAT ultimately rewards test takers who not only know the rules but can apply them to answer questions most efficiently.
Here are the main topics we’ll cover in this chapter:
Now let’s apply the Kaplan Method to the arithmetic question you saw earlier:
This is an abstract question for which the answer choices contain variables, so picking numbers will be a very efficient strategy to use. Since you are asked to find the answer choice that yields the largest absolute value, you should pick numbers at the ends of the range of possible values for x. You will need to check all of the answer choices, so even though this is a “which of the following” question, there is no benefit to starting with (E).
Evaluate each answer choice, using both x = 2 or x = −2 (the values for x with the greatest possible absolute value), in order to determine which choice will produce the largest absolute value.
You’ll need to plug in those values of x, but first notice choices (C) and (D). 3 – x and x – 3 are negatives of each other, which means they have the same absolute value. Since the question asks for the choice that has the largest possible absolute value, you can eliminate both of these choices. Notice also that with choice (E), the absolute value will be the same if using 2 or −2, since squaring the x-term will give the same value.
Since the question tests absolute value, plug in −2 first for x in each of the remaining answer choices. Choice (A) gives 3(–2) – 1, which equals −7. The absolute value of −7 is equal to 7. Choice (B) gives (–2)2 – (–2) = 4 – (–2) = 6, which has an absolute value of 6. Choice (E) gives (–2)2 + 1 = 4 + 1 = 5, which has an absolute value of 5. You already know that choice (E) will yield the same absolute value when x = 2, so eliminate choice (E); you already found that choices (A) and (B) give a larger possible absolute value.
Try x = 2 with the remaining two answer choices. In choice (A), 3(2) – 1 = 6 – 1 = 5, and the absolute value of 5 is equal to 5. For choice (B), (2)2 – 2 = 4 – 2 = 2, and the absolute value of 2 is equal to 2. The largest absolute value you found was for choice (A), when x = −2. Therefore, (A) is correct.
Confirm that your calculations are correct and that your answer makes sense.