For reasons known only to ETS, number line problems are categorized with geometry. Such problems are basically algebra problems: they present you with some diagram or some facts about certain variables and then ask whether some equations or inequalities could be true, given certain relationships among the variables. Here is a straightforward example:
A. is greater than
B. is greater than
C. The two quantities are equal
D. It cannot be determined which, if any, is greater.
As noted below, number lines have increasing numbers as one goes towards the right. So
Now, a number line simply gives all the numbers from least to greatest. So the numbers to the right of the number line are greater than those to the left. And if there are markings on the number line, then you may (provided the question says nothing to the contrary) assume that the markings are evenly spaced. What does this mean? Well in the following diagram:
you know that the distance between and is only half of the distance between and . Using algebra, we could express this as:
Now, how do we solve number line problems? It will depend on what the question asks for. Some questions ask for which of the following could be true, whereas others ask about which of the following must be true.
If the questions asks which equations could be true, then we are looking for either a set of numbers that makes the equation true and is compatible with the diagram/given information, or we want to show that the equation is somehow inconsistent with our diagram/given information.
If the questions asks which equations must be true, then we are looking for either a set of numbers that is compatible with the diagram/given information but makes our equation false, or we want to show that the equation somehow follows from the diagram/given information.
Whether you should look for a specific example or some proof of the consistency/inconsistency of the given equation is a judgment call that you need to make in the moment. It's hard to give general principles about when to do which, but after doing some practice problems and just thinking about the equation at hand, you should develop some sense of how to decide if an equation is consistent or not. For example:
Let and . Must the following be true?
It might seem like, well, if is a bigger number then of course 1 divided by a bigger number must be smaller than 1 divided by a smaller number! If you keep the numerator the same, then the bigger the denominator, the smaller the whole fraction. So, for example, let and Of course in that case.
But now consider this possibility: what if is negative? So let be some positive number and be some negative number. It's easiest to think about this with concrete numbers, so let's try Then, we get that So the equation doesn't have to be true and the answer is no.
After doing some practice problems, you get a sense of which numbers to consider when you come to one of these problems. And by running through those numbers quickly, you can often show that some equation either can be satisfied (which answers whether the equation could be true) or can be violated (which answers whether the equation must be true).
The only real piece of advice I have here is two-fold: First, always try easy, concrete numbers. Don't just think about "one negative and one positive value." It's a lot more general to think that way, but also a lot harder to evaluate. Try, instead, or and . Nice, easy numbers to calculate. And if you want a small number, try or Second, it is easy to think about what happens if both are normal, natural numbers (e.g. 1 and 2). But also think about what happens if they are small, positive numbers (e.g. and ) or only one is positive (e.g. 1 and -1) or both are negative (e.g. -1 and -2). If the equation works (or is always violated) in all of those cases, then probably it's always true (or always false).
When it comes to proving that some equation is either always true/false given the diagram/information in the question, look for easy inferences. And if you can't see a way to prove it quickly, consider just flagging it and coming back. If you've gone through a few diverse examples and they all come out the same way, probably the equation actually is always true (or always false) and the time spent confirming that might be better spent on other problems.
1. The markings on the below number line are evenly spaced:
Which of the following must be true (select all that apply):
Note that we can express and in terms of :
And so, so that A must be false.
And as for B, which is greater than 0 since So B must be true.
And as for C, is a negative number multiplied by some positive number. So it is negative and C must be false.
2. Suppose we know and Which of the following must be true (select all that apply):
For A: consider what happens if is a very large negative number while are very small positive numbers. So try and Then, the equation is false. So A can be false and hence it is not the case that it must be true.
For B: let's try to find a counter-example. It's easiest to first think of some cases where So suppose and are both some normal positive number, like 2. Then, we would get:
But of course, we could pick a negative value of for which this would not be true. So let , for example. So B can be false as well.
For C: what if are extremely small? Suppose but Then, we get:
which is not true. So C can also be false.
3. Suppose we know Which of the following must be true (select all that apply):
A: Doesn't have to be true since all three numbers could be negative.
B. Doesn't have to be true since could be barely bigger than . For example:
C: Doesn't have to be true since all three numbers could be positive.
D. Doesn't have to be true since could be very small. For example: Or, let be positive and have be negative.
E. Doesn't have to be true since could be almost the size of . For example:
4. The markings in the below number line are equally spaced. Which of the following could be true:
Again, let's put all of our values in terms of . So we get:
Then, for A: and as is positive, we get that So A must be false (i.e. cannot be true).
B: Let's substitute for to get:
which, as must be positive, simply cannot be true.
C: and this cannot be less than since is positive.
Thus none of the above could have been true.
5. The markings in the below number line are equally spaced. Which of the following must be true:
This time, let's suppose that Then, we get:
Now, let's evaluate the above:
A: since So A must be true.
B: so B must be true.
C: so C must be true as well.