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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 , then which one of the following statements must also be true?

I.     xz < 0
II.    z < 0
III.   xyz < 0

  1. None
  2. I only
  3. III only
  4. I and II
  5. II and III
Solution
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  would then be zero).
 
Hence, we can divide both sides of the inequality  by y2:

 

Simplifying yields xz < 0

 

Therefore, I is true, which eliminates (A), (C), and (E).

 

Now, the following illustrates that z < 0 is not necessarily true:

–1 — 22 — 3 = –12 < 0

 

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





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