# GRE Math : Number Line

## Example Questions

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### Example Question #1 : Number Line

The range of the earnings for architecture graduates is , and the range of the salaries for engineering graduates is .

Which of the following statements individually provide(s) sufficient additional information to determine the range of the salaries of all graduates between the two professions?

A: The median salary for the engineers is  greater than that of the architects.

B: The average (arithmetic mean) of the engineers is  greater than that of the architects.

C: The lowest salary of the engineers is  less than the lowest of the architects.

B only

C only

A only

A, B, and C

A and C only

C only

Explanation:

The provision of the bottom-end of the engineering range is the only additional information that provides us a fixed endpoint from which we can build off of by supplementing with the ranges provided in the question, to give us the full range between both engineering and achitecture graduates. See the diagram provided to understand how this can be done.

Even if the mean and medians were provided, these additional values give us no information on the endpoints of the salaries, and the question only asks for the range.

### Example Question #2 : Number Line

What's the distance between  and  on a number line?

Explanation:

Let's draw a number line.

Since a number line is straight and contains the numbers consecutively, we just subtract  from  to get

### Example Question #3 : Number Line

Which of the following answer best fits in the picture below?

Explanation:

Open circles mean the values are excluded from the set.

The number line shows the set is between  and  exclusive.

The only value in that set would be

### Example Question #4 : Number Line

If , then where on the number line lies ?

Explanation:

Because a number line contains both positive and negative integers, we need to consider both possibilities.

is  and that value is the same as . Therefore we eliminate the  choice because  will always be greater than those values raised to the  power.

Next  is . We elminate both the positive and negative range of . If we look at the difference between  and , it's over .

Then, we should guess that  will definitely be greater than  so therefore answer is .

Remember, a negative value raised to an even power will always have a positive value.

### Example Question #5 : Number Line

If perimeter of equilateral triangle is , what is the height of the triangle?

Explanation:

Since perimeter of equilateral triangle is  and we have three equal sides, we just divide that vaue by  to get . To find height, we can set-up a proportion.

The height is opposite the angle . Side opposite  is  and the side of equilateral triangle which is opposite  is .

Cross multiply.

Divide both sides by

Let's simplify by factoring out  to get a final answer of

### Example Question #1 : How To Graph An Inequality With A Number Line

On a real number line, x1 = -4 and x2 = 14. What is the distance between these two points?

4

18

10

-18

18

Explanation:

The distance between two points is always positive. We calculate lx2 - x1l, which will give us the distance between the points.

|14- (-4)| = |14+4| = |18| = 18

### Example Question #3 : How To Graph An Inequality With A Number Line

Which of the following is a graph for the values of  defined by the inequality stated above?

Explanation:

To begin, you must simplify so that you "isolate" , (i.e. at least eliminate any coefficients from it). To do this, divide all of the members of the inequality by :

Now, this inequality represents all of the numbers between 13 and 32.  However, it does include  (hence, getting a closed circle for that value) and does not include  (hence, getting an open circle for that value). Therefore, it looks like:

### Example Question #4 : How To Graph An Inequality With A Number Line

Which of the following inequalities is represented by the number line shown above?

Explanation:

Since the inequality represents one range of values between two end points (both of which are included, given the sign being "less than or equal"), you know that whatever you answer, it must be convertible to the form:

Now, you know that it is impossible to get this out of the choices that have no absolute values involved in them. Therefore, the only options that make sense are the two having absolute values; however, here you should choose only the ones that have a , for only that will yield a range like this. Thus, we can try both of our options.

The wrong answer is simplified in this manner:

And you can stop right here, for you know you will never have  for the left terminus.

The other option is simplified in this manner:

This is just what you need!

### Example Question #6 : Number Line

Quantity A:

Quantity B:

The two quantities are equal.

Quantity B is larger.

Quantity A is larger.

The relationship cannot be determined.

Quantity B is larger.

Explanation:

It is not necessary to solve this problem by multiplying terms out. Notice that between quantities A and B, the last three terms switch places for the two large numbers. as such they can be rewritten:

Quantity A:

Quantity B:

Both quantities A and B share the exact same terms, save for two:

Quantity A:

Quantity B:

From visual inspection, it is clear that B is larger.

### Example Question #7 : Number Line

Which of the following is true?

Explanation:

Since  is always positive, and , it follows that  for all possible values.

For the A, it is possible to choose values that make the statement false, for example  and .

C is always false.

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