Given that x is a real number, is −1

Question

Given that x is a real number, is −1 < x < 3?

(1) ||x − 1| − 1| < 1

(2) (x + 1)(x − 3) < 0

gsingh0711 · Accepted Answer

Explanation:

As we know | x | = x if x > 0, and | x | = −x if x < 0.

Considering statement 1:
| | x − 1 | − 1 | < 1
⇒ −1 < | x − 1 | − 1 < 1
(If | N | < 1, then N lies between −1 and 1.)

Adding 1 to every side of the inequality ⇒ 0 < | x − 1 | < 2

Considering the LHS of equation,
| x − 1 | > 0 ⇒ x ≠ 1
(Absolute value of any number is always positive as long as the value within modulus is not zero.)

Considering the RHS of equation,
| x − 1 | < 2
⇒ −2 < x − 1 < 2
(If | N | < 2, then N lies between −2 and 2.)
⇒ −1 < x < 3

Since we are getting a definite answer from above statement , statement 1 itself is sufficient to provide the answer.

Considering statement 2:
(x + 1)(x − 3) < 0

LHS of the inequality (x + 1)(x − 3) is equal to zero if x = −1 or 3. Hence, for any value of x between −1 and 3, the LHS is negative. This can be verified by picking any value of x between −1 and 3, say x = 0. Then x + 1 is a positive value and x − 3 is a negative value. Hence, the product of x + 1 and x − 3 is negative.

Hence, statement 2 ⇒ −1 < x < 3

Since we are getting a definite answer from above statement , statement 2 itself is sufficient to provide the answer.

Answer: D.

I think, I deserve a Kudus:D Lol

prashant6923 · Answer

Explanation:

As we know | x | = x if x > 0, and | x | = −x if x < 0.

Considering statement 1:
| | x − 1 | − 1 | < 1
⇒ −1 < | x − 1 | − 1 < 1
(If | N | < 1, then N lies between −1 and 1.)

Adding 1 to every side of the inequality ⇒ 0 < | x − 1 | < 2

Considering the LHS of equation,
| x − 1 | > 0 ⇒ x ≠ 1
(Absolute value of any number is always positive as long as the value within modulus is not zero.)

| x − 1 | < 2
⇒ −2 < x − 1 < 2
(If | N | < 2, then N lies between −2 and 2.)
⇒ −1 < x < 3

Since we are getting a definite answer from above statement , statement 1 itself is sufficient to provide the answer.

If X is not equal to 1,how come x lies between -1 and 3?

faltan · Answer

I agree on that.
 gsingh0711

The statement1, "1" should be excluded to say it's sufficient. Statement 2 is sufficient, thus correct answer should be B.

parijit · Answer

Bunuel chetan2u VeritasKarishma could you comment here?

Mo2men · Answer

Dear  IanStewart  
  Bunuel

In statement 1, when x = 1, then the inequality is incorrect. Hence, the solution will be -1<x<1 & 1<x<3. Should statement 1 be sufficient? I doubt because the question includes x=1 in the range −1 < x < 3, and so should be insufficient. What do you think?

Can you help please?

Thanks in advance

IanStewart · Answer

Yes, it should definitely be sufficient. It might be easier to see why with a 250-level example question: Is x > 0? 1. x = 5 2. x > 3 If you see a question like this, the answer is D. The question is not asking "can x be equal to any positive number at all?" The question is asking "can you be certain that x is a positive number?" Using Statement 1, x clearly cannot be equal to *any* positive number at all -- x is exactly equal to 5. But that's certainly sufficient, because you can be absolutely sure that x is positive. Similarly, using Statement 2, there are positive values, like 1.5 or 2, that x cannot equal. But Statement 2 is also sufficient, because you can be completely certain that x > 0 is true. Back to the original question - if you can be completely certain that x is somewhere between -1 and 3, you have sufficient information. It's irrelevant if x cannot equal 1, say. If a statement told you "x = 2", or "0 < x < 0.5", that would be sufficient: you would know with absolute certainty that x is somewhere between -1 and 3.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

Given that x is a real number, is −1 < x < 3? (1) | |x − 1| − 1 | < 1

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions