GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 08 Dec 2019, 01:02 ### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # If x is positive, is x > 3 ? (1) (x - 1)^2 > 4 (2) (x - 2)^2 > 9

Author Message
TAGS:

### Hide Tags

Intern  B
Joined: 19 Apr 2018
Posts: 8
Re: If x is positive, is x > 3 ? (1) (x - 1)^2 > 4 (2) (x - 2)^2 > 9  [#permalink]

### Show Tags

Bunuel

Hi Bunuel,

I solved the question in the same manner as you did; however, the only difference was I kept the ">" sign for the -1 root as well. So for the first statement the solution I got was x > -1 and x > 3. This is sufficient since the overlap of the two roots is for all the values of X greater than 3. My question is about the statement in bold below. Why does the ">" indicate that the solution lies to the left of the smaller root? Is this a certain theory or concept? I wasn't aware of this so when I solved the inequality I kept the ">" sign for both roots.

(1) (x - 1)^2 > 4 --> (x+1)(x−3)>0(x+1)(x−3)>0 --> roots are -1 and 3. Now, ">" sign indicates that the solution lies to the left of a smaller root and to the right of the larger root: x<−1x<−1 or x>3x>3. Since given that xx is positive then only one range is valid: x>3x>3. Sufficient.

(2) (x - 2)^2 > 9 --> (x+1)(x−5)>0(x+1)(x−5)>0 --> roots are -1 and 5. Again, ">" sign indicates that the solution lies to the left of a smaller root and to the right of the larger root: x<−1x<−1 or x>5x>5. Since given that xx is positive then only one range is valid: x>5x>5. Sufficient.

Thank you!
Math Expert V
Joined: 02 Sep 2009
Posts: 59588
Re: If x is positive, is x > 3 ? (1) (x - 1)^2 > 4 (2) (x - 2)^2 > 9  [#permalink]

### Show Tags

Gmatstudent2018 wrote:
Bunuel

Hi Bunuel,

I solved the question in the same manner as you did; however, the only difference was I kept the ">" sign for the -1 root as well. So for the first statement the solution I got was x > -1 and x > 3. This is sufficient since the overlap of the two roots is for all the values of X greater than 3. My question is about the statement in bold below. Why does the ">" indicate that the solution lies to the left of the smaller root? Is this a certain theory or concept? I wasn't aware of this so when I solved the inequality I kept the ">" sign for both roots.

_________________
Intern  B
Joined: 19 Apr 2018
Posts: 8
If x is positive, is x > 3 ? (1) (x - 1)^2 > 4 (2) (x - 2)^2 > 9  [#permalink]

### Show Tags

Hi Bunuel,

Apologies for the follow-up question and the back and forth. I read the explanation in the links in your original post, but I'm still not 100% tracking. If the inequality is (x-3) (x+1) > 0, why is that any different than when the equation is = 0? I don't understand the reasoning behind changing the ">" sign for the root -1.

Thank you

Bunuel wrote:
Gmatstudent2018 wrote:
Hi Bunuel,

I solved the question in the same manner as you did; however, the only difference was I kept the ">" sign for the -1 root as well. So for the first statement the solution I got was x > -1 and x > 3. This is sufficient since the overlap of the two roots is for all the values of X greater than 3. My question is about the statement in bold below. Why does the ">" indicate that the solution lies to the left of the smaller root? Is this a certain theory or concept? I wasn't aware of this so when I solved the inequality I kept the ">" sign for both roots.

Intern  B
Joined: 19 Sep 2019
Posts: 25
Location: Austria
Re: If x is positive, is x > 3 ? (1) (x - 1)^2 > 4 (2) (x - 2)^2 > 9  [#permalink]

### Show Tags

Bunuel wrote:
rohitgoel15 wrote:
If x is positive, is x > 3 ?

(1) (x - 1)^2 > 4
(2) (x - 2)^2 > 9

Can someone point a mistake in my method?
(1)
Taking one of the equations:
(x - 1)^2 > 4
x^2 + 1 - 2x > 4
x^2 + 1 - 2x - 4 > 0
x^2 - 3x + 1x - 3 > 0
(x-3) (x+1) > 0
x > 3 and x > -1

while in the explanation given the ans is coming out to be x > 3 and x < -1

If x is positive, is x > 3 ?

(1) (x - 1)^2 > 4 --> $$(x+1)(x-3)>0$$ --> roots are -1 and 3. Now, ">" sign indicates that the solution lies to the left of a smaller root and to the right of the larger root: $$x<-1$$ or $$x>3$$. Since given that $$x$$ is positive then only one range is valid: $$x>3$$. Sufficient.

(2) (x - 2)^2 > 9 --> $$(x+1)(x-5)>0$$ --> roots are -1 and 5. Again, ">" sign indicates that the solution lies to the left of a smaller root and to the right of the larger root: $$x<-1$$ or $$x>5$$. Since given that $$x$$ is positive then only one range is valid: $$x>5$$. Sufficient.

Solving inequalities:
http://gmatclub.com/forum/x2-4x-94661.html#p731476 (check this one first)
http://gmatclub.com/forum/inequalities-trick-91482.html
http://gmatclub.com/forum/data-suff-ine ... 09078.html
http://gmatclub.com/forum/range-for-var ... me#p873535
http://gmatclub.com/forum/everything-is ... me#p868863

Another approach:

If x is positive, is x > 3 ?

(1) (x - 1)^2 > 4 --> since both sides of the inequality are non-negative then we can take square root from both parts: $$|x-1|>2$$. $$|x-1|$$ is just the distance between 1 and $$x$$ on the number line. We are told that this distance is more than 2: --(-1)----1----3-- so, $$x<-1$$ or $$x>3$$. Since given that $$x$$ is positive then only one range is valid: $$x>3$$. Sufficient.

(2) (x - 2)^2 > 9 --> $$|x-2|>3$$. The same here: $$|x-2|$$ is just the distance between 2 and $$x$$ on the number line. We are told that this distance is more than 3: --(-1)----2----5-- so, $$x<-1$$ or $$x>5$$. Since given that $$x$$ is positive then only one range is valid: $$x>5$$. Sufficient.

Hope it helps.

Hey Bunuel,

can you point out my flaw in logic please?

(1) (x - 1)^2 > 4

x-1>2 so x>3

or

x-1>-2 so x>-1

Of course, checking the values one sees that it doesn´t work with 1 or 2, but just by taking a look at my results I figured x isn´t necessarily greater than 3.

Did the same for statement to, there I got x>5 and x>-1 and applied the same reasoning.

Intern  B
Joined: 08 Oct 2019
Posts: 3
Re: If x is positive, is x > 3 ? (1) (x - 1)^2 > 4 (2) (x - 2)^2 > 9  [#permalink]

### Show Tags

Experts Please suggest if this method is correct. I solved it as below. Request Bunuel, Veritaskarishma to suggest.

St1 (x-1)^2>4
x^2-2x+1>4
x^2-2x>3
x(x-2)>3
So x>3 or x-2>3
X>3 or x>5 Hence St 1 sufficient

St 2 (x-2)^2>9
x^2-4x+4>9
x^2-4x>5
x(x-4)>5
x>5 or x-4>5
x>5 or x>9

st 2 sufficient.

Ans Choice D
Manager  B
Joined: 19 Feb 2019
Posts: 96
Concentration: Marketing, Statistics
Re: If x is positive, is x > 3 ? (1) (x - 1)^2 > 4 (2) (x - 2)^2 > 9  [#permalink]

### Show Tags

Hi can we solve the equations in the following way

(x-1)^2>4
So on expanding we get x^2-2x+1>4;
x^2-2x>3
x(x-2)>3
so x>3 or x>5

Is this the correct way to solve this equation?
Intern  B
Status: Professional GMAT Trainer
Affiliations: GMAT Coach
Joined: 21 Mar 2017
Posts: 10
Location: United States
GMAT 1: 760 Q50 V44 GMAT 2: 770 Q51 V44 Re: If x is positive, is x > 3 ? (1) (x - 1)^2 > 4 (2) (x - 2)^2 > 9  [#permalink]

### Show Tags

Kshah001 wrote:
Experts Please suggest if this method is correct. I solved it as below. Request Bunuel, Veritaskarishma to suggest.

St1 (x-1)^2>4
x^2-2x+1>4
x^2-2x>3
x(x-2)>3
So x>3 or x-2>3
X>3 or x>5 Hence St 1 sufficient

St 2 (x-2)^2>9
x^2-4x+4>9
x^2-4x>5
x(x-4)>5
x>5 or x-4>5
x>5 or x>9

st 2 sufficient.

Ans Choice D

Your mistake is in this step:
"x(x-2)>3
So x>3 or x-2>3"

You are most likely confusing this with when it is equal to zero -- for example:
x(x-2) = 0 ---> in this case, x = 0 OR x-2 = 0. (One of the two things we are multiplying on the left must be zero, to get zero on the right)
Important ---> This does not apply if it's not equal to zero.

Instead, we want to take the square root of both sides:

(1) $$(x - 1)^2$$ > 4
|x-1| > 2 (Important: when we take the square root of a square, we get the absolute value. It's a common mistake to forget this. For this problem, it doesn't affect the answer, because the question says "if x is positive", and the negative values don't apply. Another common mistake is to miss the "If x is positive" -- read carefully!)

Case A: If x-1 is positive, then x-1 > 2 ---> x > 3
Case B: If x-1 is negative, then the absolute value flips the signs on the left: -x+1 > 2 ---> x < -1 ---> this doesn't apply, because the question says "If x is positive".
Therefore, our answer to the question, "is x>3?", is YES, and (1) is sufficient.

We do the same process for statement 2:
(1) $$(x - 2)^2$$ > 9
|x-2| > 3
x > 5
Again, the negative case doesn't apply here. (If x-2 is negative, then the absolute value flips the signs on the left: -x+2 > 3 ---> x < -1 ---> this doesn't apply, because the question says "If x is positive")
Therefore, our answer to the question, "is x>3?", is YES, and (2) is sufficient.

_________________
6 Principles for Effective GMAT Training: https://www.gmatcoach.com/our-approach/

I offer online tutoring -- please visit gmatcoach.com/trial to request a free consultation. Re: If x is positive, is x > 3 ? (1) (x - 1)^2 > 4 (2) (x - 2)^2 > 9   [#permalink] 07 Dec 2019, 21:48

Go to page   Previous    1   2   [ 27 posts ]

Display posts from previous: Sort by

# If x is positive, is x > 3 ? (1) (x - 1)^2 > 4 (2) (x - 2)^2 > 9  