## Example Questions

### Example Question #1 : Graphing An Inequality With A Number Line

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

-18

10

4

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 #2 : Graphing 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 #3 : Graphing 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: Quantity A is larger.

The two quantities are equal.

The relationship cannot be determined.

Quantity B is larger.

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.

### Example Question #61 : Integers    Which of the statements is always true?      Explanation:

Although C may approach zero for large values of and values of , it will never actually reach it for real values.

A and B, however, have x and y values which make them false:

For  A, an example is For B, would make the value zero.

### Example Question #62 : Integers Which of the following is a graph for the values of defined by the inequality stated above?      Explanation:

Begin by solving for :   Now, this is represented by drawing an open circle at 6 and graphing upward to infinity:  