**Positive & Negative Numbers**

A number greater than 0 is positive. On the number line, positive numbers are to the right of 0. A number less than 0 is negative. On the number line, negative numbers are to the left of 0. Zero is the only number that is neither positive nor negative; it divides the two sets of numbers. On the number line, numbers increase to the right and decrease to the left.

The expression x > y means that x is greater than y. In other words, x is to the right of y on the number line:

We usually have no trouble determining which of two numbers is larger when both are positive or one is positive and the other negative (e.g., 5 > 2 and 3.1 > â€“2). However, we sometimes hesitate when both numbers are negative (e.g., â€“2 > â€“4.5). When in doubt, think of the number line: if one number is to the right of the number, then it is larger. As the number line below illustrates, â€“2 is to the right of â€“4.5. Hence, â€“2 is larger than â€“4.5.

**Miscellaneous Properties of Positive and Negative Numbers**

- The product (quotient) of positive numbers is positive.
- The product (quotient) of a positive number and a negative number is negative.
- The product (quotient) of an even number of negative numbers is positive.
- The product (quotient) of an odd number of negative numbers is negative.
- The sum of negative numbers is negative.
- A number raised to an even exponent is greater than or equal to zero.

**Example:**

If *x**y*^{2}*z* < 0 , then which one of the following statements must also be true?

I. *xz* < 0

II. *z* < 0

III. *xyz* < 0

(A) None

(B) I only

(C) III only

(D) I and II

(E) II and III

Since a number raised to an even exponent is greater than or equal to zero, we know that y^{2} is positive (it cannot be zero because the product *x**y* ^{2}*z* would then be zero). Hence, we can divide both sides of the inequality *xy*^{2}*z* < 0 by *y*^{2}:

$\frac{x{y}^{2}z}{{y}^{2}}<\frac{0}{{y}^{2}}$

Simplifying yields xz < 0

â€‹

Therefore, I is true, which eliminates (A), (C), and (E). Now, the following illustrates that *z* < 0 is not necessarily true:

â€“1 â€¢ 2^{2} â€¢ 3 = â€“12 < 0

This eliminates (D). Hence, the answer is (B).