Solutions

1. Must Be True

   Although you don’t have specific values for either s or q, you know that s is greater than 1, and you know that q is between 0 and −1. Even if s was as small as it could be (≈ 1.00001) and q was as negative as it could be (≈ −0.99999), the sum would still be positive.

2. Could Be True

   The product pq will be positive since both p and q are negative. You know that t must be between 0 and 1, but the product pq could be either less than 1 or greater than 1, depending on the numbers chosen. If t = 0.9, q = −0.1, and p = −2, then t > pq. However, if t = 0.5, q = −0.9, and p = −5, pq > t.

3. Could Be True

   Both p² and s⁴ will be positive, but depending on the numbers chosen for p and s, either value could be larger. If p = −2 and s = 3, then s⁴ > p². If p = −8 and s = 2, then s⁴ < p².

4. Must Be True

   You know s is greater than 1 and p is less than −1. The smallest that the difference can be is greater than 2.

   You know r must be between 0 and 1 and q must be between 0 and −1. The greatest the difference can be is less than 2. Thus, s − p will always be greater than r − q.

5. Never Be True

   Even if t is as large as it can be and q is as small as it can be, the difference will still have to be less than 2. If t = 0.999999 and q = −0.999999, then t − q = 1.999998.

6. Could Be True

   If r = 0.1 and s = 2, then rs < 1. If r = 0.5 and s = 3, then rs > 1.

7. 2.5

   To figure out the distance between Y and Z, you first need to figure out the distance between tick marks. You can use the two points on the number line to find the distance. There are 7 intervals between the two points, as shown here:

   

   

   You actually do not need to know the positions of Y and Z to find the distance between them. You know that there are 4 intervals between Y and Z, so the distance is:

   
8. (A)

   The trick to this problem is recognizing that there is more than one possible arrangement for the points on the number line. Because is longer than , A could be either in between B and C or on one side of B with C on the other side of B, as shown here:

   

   Using the information about the midpoints (D and E) and the lengths of the line segments, you can fill in all the information for the two number lines:

   

   You can see that has two possible lengths: 1 and 9, however 1 is the only option that is an answer choice.

9. (D)

   With only one actual number displayed on the number line, you have no way of knowing the distance between tick marks. If the tick marks are a small fractional distance away from each other, then will be greater than −1. For instance, if the distance between tick marks is , then A is , B is and is , which is greater than −1. If the distance between tick marks is 1, then A is −2, B is 4, and is −8, which is less than −1.

   Therefore, the relationship cannot be determined from the information given.

10. (A)

    The easiest approach is to pick numbers. Point q must be a negative number and r must be positive. If q = −1, then r = 2:

    

    If s is the midpoint of q and r, then s must be 0.5. Therefore, s > 0.

    For any numbers you pick, s will be positive. Therefore, Quantity A is greater.

11. (C)

    

    Visualizing the preceding number line, the ratio of to is . Similarly, the ratio of to is . Therefore, the two quantities are equal.