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

1. The product (quotient) of positive numbers is positive.
2. The product (quotient) of a positive number and a negative number is negative.
3. The product (quotient) of an even number of negative numbers is positive.
4. The product (quotient) of an odd number of negative numbers is negative.
5. The sum of negative numbers is negative.
6. A number raised to an even exponent is greater than or equal to zero.

Example:

If xy2z < 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 y2 is positive (it cannot be zero because the product xy 2z would then be zero). Hence, we can divide both sides of the inequality xy2z < 0 by y2:

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

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

â€“1 â€¢ 22 â€¢ 3 = â€“12 < 0

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