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GRE Quantitative Practice, Complex Multiple-Choice

Question 3

If a > 0 > b > c, which of the following inequalities holds in all cases?

Indicate all such inequalities.

A.  ab + c < 0

B.  bca < 0

C.  ac + b < 0

Explanatory Answer

Here's an analysis of the three answer choices:

(A) The product of a (a positive number) and b (a negative number) must be negative (less than zero). Adding c (a negative number) to that product always yields a negative number further to the left on the number line. Inequality (A) holds in all cases.

(B) Since b and c are both negative (less than zero in value), their product must be a positive number. But whether that product is greater than a depends on the specific values of the three numbers. For example, if b = –1, c = –2 and a = 3, then bca < 0. But if b = –2, c = –3 and a = 4, then bca > 0. These two examples suffice to show that inequality (B) does not hold in all cases.

(C) The analysis of inequality (C) is the same as the analysis of inequality (A). The product of a (a positive number) and c (a negative number) must be negative (less than zero). Adding b (a negative number) to that product always yields a negative number further to the left on the number line. Inequality (C) holds in all cases.

The correct answer is (A) and (C). You would gain credit for answering the question correctly only by selecting these, and only these, two choices.