Problem Solving

If b < e < y+k and b-1 < w < k, then which of the following MUST be true?

I. 2b-1 < w+e < y+2k
II. 1 < e-w < y
III. b < k+2

(A) I only
(B) III only
(C) I and II only
(D) I and III only
(E) II and III only

The OA is the option D.

Why II is not true? And, how can I show that III is true? I know how to prove that I is true. Experts, can you help me? Thanks.

Hello Vjsesus12.

Let's take a look at your question.

We have the following inequality: $$b<e<y+k\ \ and\ \ b-1<w<k\ .$$ Adding both inequalities we will get $$2b-1<e+w<y+2k.$$ Hence, I is true..

Now we cannot subtract the inequalities, so II is not true. Let's consider the following counter-example: b = -10, e = -5, y = 0, k = 4 and w = 0.
These values satisfy both of the given inequalities (b < e < y+k and b-1 < w < k), however, when we plug these values into statement II (1 < e-w < y), we get: 1 < -5 < 0, which is not true.

Finally, from the first inequality we have $$b-1<k\ \Rightarrow\ \ b<k+1,\ and\ k+1<k+2\ \Leftrightarrow\ b<k+2.$$ Therefore, III is true.

In conclusion the correct answer is D.

I hope this explanation may help you.

I'm available if you'd like a follow-up.

Regards.