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Inequalities problem

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by Troika » Mon Aug 08, 2011 10:08 am
If d>0 and 0<1-c/d<1, which of the following must be true?

I. c>0

II. c/d<1

III. c^2+d^2>1

A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III

OA: C
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by Troika » Tue Aug 09, 2011 2:22 am
Thank you Mitch!
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by goalevan » Wed Aug 10, 2011 6:05 pm
d > 0 and 0 < 1 - c/d < 1, so:

0 < (d - c)/d < 1
0 < d - c < d (we can multiply and preserve the direction of inequality since d > 0)
c < d < c + d
If c + d > d, then c + d - d > d - d, or c > 0.

Which of the following must be true?

I. c > 0. check, this must be true as shown above.

II. c/d < 1. c < d and d is positive as shown above, so c/d < d/d = 1. check

III. c^2 + d^2 > 1. (c,d) could be (0.000000001, 0.0001), which is less than 1, or could be (100000, 10000000000000), which is greater than 1. This does not have to be true.
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