Bunuel wrote:
If d > 0 and 0 < 1 - c/d < 1, which of the following must be true?
I. c > 0
II. c/d < 1
III. c² + d² > 1
(A) I only
(B) II only
(C) I and II only
(D) II and III only
(E) I, II and III
Take the inequality: 0 < 1 - c/d < 1
Subtract 1 from all sides: -1 < -c/d < 0
Multiply all sides by -1 to get: 1 > c/d > 0
[when we multiply an inequality by a NEGATIVE value, we must REVERSE the direction of the inequality symbols]Rearrange to get: 0 < c/d < 1
We can immediately see that statement II (c/d < 1) is true
Also, since 0 < c/d, we know that c/d is positive.
Since we're told d is positive, we can conclude that
c must also be positiveIn other words, c > 0, which means statement I is true.
Now let's analyze statement III
Notice that the values c = 0.1 and d = 0.2 satisfy all of the given information.
Now recognize that c² + d² = 0.1² + 0.2² = 0.01 + 0.04 = 0.05
So, it's not necessarily the case that c² + d² > 1
In other words, statement III need not be true.
Answer: C
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