If xâ‰ 0, is |x| < 1?

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VJesus12: Legendary Member; Posts: 2276; Joined: Sat Oct 14, 2017 6:10 amFollowed by: 3 members

by VJesus12 » Mon May 28, 2018 1:36 am

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If xâ‰ 0, is |x| < 1?

(1) x^2 < 1
(2) |x| < 1/x

The OA is the option D.

Why is sufficient the second statement? How can I prove it? Can someone help me? Thanks.

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Source: Beat The GMAT — Data Sufficiency | Mon May 28, 2018 1:36 am

Vincen: Legendary Member; Posts: 2898; Joined: Thu Sep 07, 2017 2:49 pmFollowed by: 5 members; Thanked: 6 times

by Vincen » Mon May 28, 2018 2:47 am

VJesus12 wrote:If xâ‰ 0, is |x| < 1?

(1) x^2 < 1
(2) |x| < 1/x

The OA is the option D.

Why is sufficient the second statement? How can I prove it? Can someone help me? Thanks.

Hello Vjesus12.

Let's see your question.

(1) x^2 < 1

This statement implies that $$\sqrt{x}<\sqrt{1}\ \ \ \Leftrightarrow\ \ \ \left|x\right|<1.\ $$ Hence, this statement is sufficient.

(2) |x| < 1/x

Since xâ‰ 0, then |x|>0.

Now, if 0 < |x| < 1/x, then it implies that x must be positive.

If x>0, then |x|=x. Therefore, $$\left|x\right|<\frac{1}{x}\ \ \Rightarrow\ \ x<\frac{1}{x}\ \ \ \Rightarrow\ \ x\cdot x<1\ \ \ \Rightarrow\ \ \ x^2<1.$$ This last expression is the same as the first statement, therefore, |x|<1. This statement is also sufficient..

Therefore, the correct answer for this DS question is the option D.

I hope it helps.

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