SOLUTION: Which of the following statements must be true when a^2 < b^2 and a and b are not 0? I. a^2/a < b^2/a II. 1/a^2 > 1/b^2 III. (a + b) (a

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Question 308661: Which of the following statements must be true when a^2 < b^2 and a and b are not 0?
I. a^2/a < b^2/a
II. 1/a^2 > 1/b^2
III. (a + b) (a - b) < 0

Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!
I. Clearly this is false. If , then the inequality sign should flip, but it does not. If the sign did flip, then it would only be true for negative values of 'a', but what if 'a' was positive? Since we have this uncertainty, statement I is false.

II. If and , then and are both positive numbers. Recall that if and 'x' and 'y' are both positive, then which shows us that statement II is true.

III.

Start with the given inequality.

Subtract from both sides.

Factor the left side using the difference of squares.

So statement III is true.

SOLUTION: Which of the following statements must be true when a^2 < b^2 and a and b are not 0? I. a^2/a < b^2/a II. 1/a^2 > 1/b^2 III. (a + b) (a - b) < 0