M14-13

Question

If  x  is an integer, is  |x| \gt 1 ?

(1)  (1 - 2x)(1 + x) \lt 0 
 
(2)  (1 - x)(1 + 2x) \lt 0

Bunuel · Accepted Answer

Official Solution: If x is an integer, is |x| \gt 1 ? Is |x| \gt 1 ? Is x < -1 or x > 1 ? So, for an YES answer, x can be any integer except -1, 0, and 1. (1) (1 - 2x)(1 + x) \lt 0 . Rewrite as (2x - 1)(x + 1) > 0 (so that the coefficient of x^2 is positive after expanding): the roots are x=-1 and x=\frac{1}{2} . The " > " sign indicates that the solution lies to the left of the smaller root and to the right of the larger root; therefore, the given inequality holds true for: x < -1 and x > \frac{1}{2} . However, x could still equal 1, so this statement is not sufficient. (2) (1 - x)(1 + 2x) \lt 0 . Rewrite as (x - 1)(2x + 1) > 0 : the roots are x=-\frac{1}{2} and x=1 . The " > " sign indicates that the solution lies to the left of the smaller root and to the right of the larger root; therefore, the given inequality holds true for: x < - \frac{1}{2} and x > 1 . However, x could still equal -1, so this statement is not sufficient. (1)+(2) The intersection of the ranges from (1) and (2) is x < -1 and x > 1 . This is sufficient to determine that |x| > 1 . P.S. For a more detailed explanation of the approach used to solve this question, please refer to the following link: Solving Quadratic Inequalities: Graphical Approach Answer: C

Cadaver · Answer

I did by wavy curve method.....Efficient

TheMechanic · Answer

Hi Bunuel,

I know this has to do something with inequalities changing signs when you go beyond extremes. But I needed a refresher on that. In particular I am stuck with how did you come up with this bit:

Is there any post which I can refer to solidify my concept of finding valid extreme values from Roots of the equation?

Thanks in advance.

Vaibhav.

Ok after doing a bit of looking around, I found this:

https://gmatclub.com/forum/inequalities-trick-91482.html#p804990

This serves the purpose.

Bunuel, is there any other slick/intuitive way of doing this such that one saves time without drawing the line graph and plotting roots on it during the exam?

Bunuel · Answer

Check below links:

Inequalities Made Easy!

Solving Quadratic Inequalities - Graphic Approach 
 Inequality tips

DS Inequalities Problems  
 PS Inequalities Problems

700+ Inequalities problems

NeverGiveUp- Arpit · Answer

Hi  Bunuel ,

My approach is a bit different from yours but uses the same concept. But I'm getting an incorrect answer. Could you please advise which step I'm following wrong?

As per question Stem -

x >1
X< -1

Choice A ( I didn't multiply by -1 the way you did)

I got equation - -1< X <1/2 - So answer is No for the value of X

Choice B ( I didn't multiply by -1 the way you did)

Equation : -1/2 < X < 1 - So answer is No for the value of X

Hence I marked D in GMAT Club test which is incorrect. Could you please advise how we can solve without multiplying by - 1? And in which step I'm wrong?

niks18 · Answer

Hi @NeverGiveUp- Arpit The highlighted portion is incorrect. even if you are not multiplying with -1, then you need to realize that co-efficient of x will be negative, so on a number line range will start from negative values. Hence when you plot the roots of the quadratic equation on a number line then the range of x will be in the negative regions because the inequality is negative. Refer below image for clarity - inequality.jpg Similarly for statement B basically as per your range 0 is a possible solution. so when you put x=0 in statement 1, you will get 1<0, which is not possible

dcummins · Answer

Bunuel , thanks for the explanation.

I was able to get the same answers during the test I did, but I was really confused by the two ranges given.

I was wondering if you can clarify 2 points for me please.

Point 1: The question asks is |x|>1 ? so I thought we had to prove whether x>0 and -x>0 (translating to...) --> x<1
As in, we need to ensure both x>1 and x<1 holds true

Is my interpretation correct?

Point 2:
I'm not sure how you came to a conclusion based on the "intersection" as there are two potential ranges

x>1/2 and x<-1 (from statement 1); and
x>-1/2 and x>1 (from statement 2)

Why did you take x<-1 and x>1 as sufficient to rule out the other range (x>1/2 and x>-1/2)?

Bunuel · Answer

1. The question asks whether |x| > 1, not whether |x| > 0.

|x| > 1 means x < -1 or x > 1.
While |x| > 0, holds true for ANY value but 0.

2. From (1):  x \lt -1 and  x \gt \frac{1}{2} 
  ----------- (-1)--------------------------------(1/2) ------------------------

From (2):  x \lt - \frac{1}{2}  and  x \gt 1 .
  ------------------------ (-1/2)------------------------------------(1) ------------------------

Ranges which satisfy BOTH are:
  ----------- (-1)----------------------------------------------------(1) ------------------------

VeritasPrepErika · Answer

Quick and easy method of solving if the inequalities are giving you grief:

The question asks if  |x| > 1 . Rewriting this question, we get "is  |x| > 1  OR  |x| < -1 . Given that x is an integer, the only values that will give us a "no" for this question are -1, 0, and 1. All other integer values will give us a "yes".

Knowing this, we can try to prove each statement insufficient by getting both a "no" and a "yes".

Statement 1 tells us that  (1-2x)(1+x)  is negative. To get a "yes", we can plug in an easy number, like 2 — this gives  (1-4)(1+2) = (-3)(3) = -9 , which is negative. This works, so we get a "yes". To get a "no", we can try -1, 0, or 1. Since 2 worked, 1 might also be a good option — this gives  (1-2)(1+1) = (-1)(2) = -2 , which is negative. This works, so we get a "no". Since we got both a "no" and a "yes", this Statement is insufficient.

Statement 2 tells us that  (1-x)(1+2x)  is negative. Let's try the same numbers again. Plugging in 2 gives  (1-2)(1+4) = (-1)(5) = -5 , which is negative. This works, so we get a "yes". Plugging in 1 gives  (1-1)(1+2) = (0)(3) = 0 , which NOT negative. That didn't work, so we should try one of our other "no" options. Plugging in 0 gives  (1-0)(1+0) = (1)(1) = 1 , which NOT negative. That didn't work either, so we're on to our last "no" option. Plugging in -1 gives  (1-(-1))(1+(-2)) = (2)(-1) = -1 , which is negative! This works, so we get a "no". Since we got both a "no" and a "yes", this Statement is insufficient.

Putting both statements together: the two statements both work with x=2, so we can get a "yes" with both statements. The only "no" answer we could get for Statement 2 was x=-1. If x=-1 works with Statement 1 as well, we will get a "no" with both statements, but if it doesn't, there is no way to get a "no" with both statements. Plugging -1 into Statement 1 gives  (1-(-2))(1+(-1)) = (3)(0) = 0 , which is not negative. This doesn't work, so we can't get a "no" with both statements! This means that the answer can only be "yes" with both statements, and the answer is C.

When you have limited options for what will give you a "yes" or a "no" in a yes/no Data Sufficiency question, you can work directly with those options!

danklin · Answer

I think this is a  high-quality  question and I agree with explanation.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

M14-13

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