Solving Math Problems with Number Lines on the SAT

In this lesson, you'll get an overview of what number lines are, some tips for tackling them on the SAT and a preview of the kinds of number line problems on the test.

Number Lines

A number line is a straight line on which every real number is represented by one point. Number lines show the relative sizes of all the numbers on the line: numbers to the left are progressively smaller, and numbers to the right are progressively greater. In elementary school, number lines were there to help you solve problems - but on the SAT, they work a little differently. The number line isn't part of the solution, it's part of the problem, and you're the one in charge of figuring out what on Earth it all means.

The heart of a number line problem is the relationship among all the quantities on the line - this is what you'll have to manipulate in various ways to solve the problems correctly. In this section, we'll take a look at how that works, what kind of number line problems you'll see on the test and how to interpret them.

SAT Number Lines

You're probably used to number lines that look something like this.

Standard number line

One line, a bunch of numbers, usually with zero in the middle. Sometimes the scale will be different, but the pattern is generally the same. That's the classic number line. But on the SAT, the number lines look more like this.

SAT style number line

One line, a bunch of numbers and some variables inserted as the basis of the question. Some of them even look like this. No numbers at all, just variables floating around in space. These tick marks could be units of one or units of a thousand; you have no way to know.

And just to make it even more complicated, the tick marks on the number line aren't always the same distance apart; you'll have to pay very close attention to the individual question to avoid making unjustified assumptions about the picture.

Example Question

So, what kind of questions are you likely to see on a number line? How does the SAT take such a simple tool and come up with problems that can stump high school students? Here's an example:

On the number line shown, the tick marks are evenly spaced. Which of the following must be true of x?

1. X is between 0 and 1
2. X is between -1 and 0
3. X is between 1 and 2
4. X equals -1
5. X equals 1

You can see how this one is a souped-up version of the 'variables in place of numbers' theme. Now let's take a look at how we'll solve it.

Example Solution

Remember that the core of a number line question is the relationship among the numbers. In this case, we have no absolute standard: there are no real numbers on the line. So we'll use inequalities to get a handle on the relationships shown. Let's start on the left. We can see that 2x is less than x because it is to the left of x on the number line.

This tells us that whatever x is, it will be greater than 2x. This is great: it lets us ignore the number line for a minute and just work with this equation. Any answer choice that doesn't satisfy this inequality is automatically out. So, we'll plug them in and see.

We'll start with answer choice (A). Pick a number between 0 and 1 as a test. Let's take 0.5. If you multiply it by 2, do you get a smaller number?

Nope. 0.5 times 2 is 1, which is bigger. So, cross off (A). You don't need to test every number in the group because the question asks you which of the statements in the answers must be true. If even one number in the range is false, the whole answer is bunk.

Now we'll move on to (B). For a number between -1 and 0, we'll try -0.5. Multiply -0.5 by 2, and we get -1. That checks out, so we'll keep (B) for now.

What about (C)? If we multiply 1.5 by 2, we get 3, which is bigger than 1.5. Cross it off and move on.

In choice (D), we see that -1 times 2 is -2: smaller than -1, so that definitely works. For (E), multiplying 1 times 2 gives us 2, which is bigger than 1, so this is an easy elimination.

Now we're already down to just two answer choices left. To choose between them, we'll have to look back at the number line again. We can see from the line that the absolute value of x is greater than x squared. Now we can forget about the three answers we already crossed off, and just focus on the two we have left.

If we try to plug x equals -1 into this, we get 1 is greater than 1. This is obviously false. But if we try plugging in -0.5, we get 0.5 is greater than 0.25, which is correct. So, we know by elimination that answer choice (B) is correct.

If abstract algebra is more your style, you can also use that to check your answer. If x is greater than 2x, then xmust be negative. So, you can cross off (A), (C) and (E) just like that. And if the absolute value of x is greater than x squared, then x must be a fraction because fractions are the only numbers that get smaller in absolute value when you square them. So (D) can't possibly fit. But even if you're not comfortable with the abstract math, you can still figure all of this out by plugging in points and testing to see what satisfies both conditions; you don't have to intuit the answer.