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 Your Progress

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.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 500,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

Can you explain what was wrong with the way I was approaching the problem?

I mean other than the part that you marked red, what was I doing wrong? Do I have to solve theproblem using the solution that you mentioned?

If x > -2, how is |x + 2| = (x - 2)? Is there an identity that I am missing? If I plug in, X = -1, |x + 2| = 1, but (x - 2) = -3

Why the discrepancy? What identity am I missing?

There was a typo: When \(x\leq{-2}\), then \(|x+2|=-(x+2)\) When \(x>{-2}\), then \(|x+2|=(x+2)\).

Absolute value properties:

When \(x\leq{0}\) then \(|x|=-x\), or more generally when \(some \ expression\leq{0}\) then \(|some \ expression|={-(some \ expression)}\). For example: \(|-5|=5=-(-5)\);

When \(x\geq{0}\) then \(|x|=x\), or more generally when \(some \ expression\geq{0}\) then \(|some \ expression|={some \ expression}\). For example: \(|5|=5\).

For our question, when x>-2 (when x+2>0), |x+2|=x+2.

Hope it's clear.

Got it.

Thanks,

Final question, why are there two possibilities for when x = 0? Is that correct? or a typo?

No, that's not a typo: |0|=0=-0.
_________________

x<1 x^2 - 4x + 6 = 3 - |x - 1| x^2 - 4x + 6 = 3 - -(x-1) x^2 - 4x + 6 = 3 - (-x+1) x^2 - 4x + 3 = + x - 1 x^2 - 5x + 4 = 0 (x-1)(x-4) = 0 x=1, x=4 Neither 1 or 4 fall within the range of x<1 INVALID

The product is (2)

(C)

(Bunuel, could you explain to me how we know where the greater than or equals sign goes in these problems? I see that in your solution you had x>=1, but I don't know why that is as opposed to say, x<=1)

Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]

Show Tags

19 Jun 2014, 02:57

Bunuel wrote:

carcass wrote:

What is the product of all the solutions of x^2 - 4x + 6 = 3 - |x - 1| ?

(A) -8 (B) -4 (C) 2 (D) 4 (E) 8

If \(x<1\), then \(|x - 1| = -(x-1)=1-x\), so in this case we'll have \(x^2 - 4x + 6 = 3-(1-x)\) --> \(x^2-5x+4=0\) --> \(x=1\) or \(x=4\) --> discard both solutions since neither is in the range \(x<1\).

If \(x\geq{1}\), then \(|x - 1| = x-1\), so in this case we'll have \(x^2 - 4x + 6 = 3-(x-1)\) --> \(x^2-3x+2=0\) --> \(x=1\) or \(x=2\).

Therefore, the product of the roots is 1*2=2.

Answer: C.

Dear Bunuel,

We do not need to consider two situation of |x - 1|.

What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]

Show Tags

11 Aug 2015, 18:03

paul16 wrote:

Hello everyone,

I am so close to understanding this question, but the one thing I do not understand is why the positive of |x+2| is >= and the negative of |x+2| is just <?

Sorry if its a dumb question

Paul

I have this question too. Typically it doesnt matter if we use the +ve or -ve absolute value function since the value is zero anyways. But for this case, we need to use the value to include/exclude values, so is >= and < the standard?

I am so close to understanding this question, but the one thing I do not understand is why the positive of |x+2| is >= and the negative of |x+2| is just <?

Sorry if its a dumb question

Paul

I have this question too. Typically it doesnt matter if we use the +ve or -ve absolute value function since the value is zero anyways. But for this case, we need to use the value to include/exclude values, so is >= and < the standard?

1 thing, the text in red is not always true. |x| can be any value other than 0 as well.

Coming back to your question, the reason we include "=" with ">" sign because |x| = x for x\(\geq\)0 while |x| =-x for x<0

The convention is to always include "+" with ">" and not with "<" as the NATURE of |x| remains the same for x\(\geq\) 0, while the nature changes for |x| (we need to put a negative infront of 'x') when x<0.

For example, |4| = 4 or |0| = 0 but |-5| = -(-5) = 5

\(x+2=0\) --> \(x=-2\); OR \((x+2)^2-1=0\) --> \((x+2)^2=1\) --> \(x=-1\) or \(x=-3\).

The product of the roots: \((-2)*(-1)*(-3)=-6\).

Answer: A.

Hope it's clear.

Hi Bunuel,

When I tried to follow the same method I get different answer for the below question.

x^2 + 4x + 7 = |x + 2| + 3.

When x <= 0 then |x + 2| = - (x+2 ) = -x - 2.

Then x^2 + 4x + 7 = - x - 2 + 3.

=> x^2 + 4x + 7 = - x + 1. => x^2 + 5x + 6 = 0 => ( x +2 ) ( x + 3) = 0 => x = -2 and - 3 . ( Since x < = 0 both the numbers are possible ).

When x > 0 then |x + 2| = x + 2.

Then x^2 + 4x + 7 = x + 2 + 3. => x^2 + 4x + 7 = x + 5. => x^2 + 3x + 2 = 0 . =>( x + 2 ) ( x + 1 ) = 0. x = - 2 and - 1 ( Since x > 0 both numbers doesn't fit in the range ).

Then suitable numbers would be -2 and -3 and the product should be 6.

\(x+2=0\) --> \(x=-2\); OR \((x+2)^2-1=0\) --> \((x+2)^2=1\) --> \(x=-1\) or \(x=-3\).

The product of the roots: \((-2)*(-1)*(-3)=-6\).

Answer: A.

Hope it's clear.

Hi Bunuel,

When I tried to follow the same method I get different answer for the below question.

x^2 + 4x + 7 = |x + 2| + 3.

When x <= 0 then |x + 2| = - (x+2 ) = -x - 2.

Then x^2 + 4x + 7 = - x - 2 + 3.

=> x^2 + 4x + 7 = - x + 1. => x^2 + 5x + 6 = 0 => ( x +2 ) ( x + 3) = 0 => x = -2 and - 3 . ( Since x < = 0 both the numbers are possible ).

When x > 0 then |x + 2| = x + 2.

Then x^2 + 4x + 7 = x + 2 + 3. => x^2 + 4x + 7 = x + 5. => x^2 + 3x + 2 = 0 . =>( x + 2 ) ( x + 1 ) = 0. x = - 2 and - 1 ( Since x > 0 both numbers doesn't fit in the range ).

Then suitable numbers would be -2 and -3 and the product should be 6.

Please clarify me if I am missing anything...

The transition point (value) must be the point for which the expression in modulus changes its sign. For |x + 2| it's -2 not 0. So, you should consider the ranges when x<=-2 and when x>-2. Check highlighted text in my solution.
_________________

Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]

Show Tags

25 Jul 2016, 20:52

Bunuel wrote:

carcass wrote:

What is the product of all the solutions of x^2 - 4x + 6 = 3 - |x - 1| ?

(A) -8 (B) -4 (C) 2 (D) 4 (E) 8

If \(x<1\), then \(|x - 1| = -(x-1)=1-x\), so in this case we'll have \(x^2 - 4x + 6 = 3-(1-x)\) --> \(x^2-5x+4=0\) --> \(x=1\) or \(x=4\) --> discard both solutions since neither is in the range \(x<1\).

If \(x\geq{1}\), then \(|x - 1| = x-1\), so in this case we'll have \(x^2 - 4x + 6 = 3-(x-1)\) --> \(x^2-3x+2=0\) --> \(x=1\) or \(x=2\).

Therefore, the product of the roots is 1*2=2.

Answer: C.

Hi Bunuel,

I don't know whether my alternative method is right to solve this question:

1) I tried to create a square: X^2-4X+6=3-|X-1| X^2-4X+3=-|X-1| X^2-2X+1-2X+2=-|X-1| (X-1)^2-2(X-1)+|X-1|=0

2) replace X-1 with Y Y^2-2Y+|Y|=0

3) if Y>0, the equation will be: Y^2-Y=0, Y(Y-1)=0, then X-1=0 or X-1=1 So we have X=1 or X=2

4) if Y=0, X-1=0, X=1

4) if Y<0, all in the left side are positive, thus cannot resulting in 0

Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]

Show Tags

25 Jul 2016, 21:21

mod can be taken + and -ve when taken +ve , solving the equation will give roots 1 and 2 , product will be 2 when taken -ve , solving equation will give 1 and 4 , which can be discarded as we have considered x and -ve and we are getting roots positive. answer will be 2

Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]

Show Tags

10 Oct 2016, 08:47

Bunuel wrote:

carcass wrote:

What is the product of all the solutions of x^2 - 4x + 6 = 3 - |x - 1| ?

(A) -8 (B) -4 (C) 2 (D) 4 (E) 8

If \(x<1\), then \(|x - 1| = -(x-1)=1-x\), so in this case we'll have \(x^2 - 4x + 6 = 3-(1-x)\) --> \(x^2-5x+4=0\) --> \(x=1\) or \(x=4\) --> discard both solutions since neither is in the range \(x<1\).

If \(x\geq{1}\), then \(|x - 1| = x-1\), so in this case we'll have \(x^2 - 4x + 6 = 3-(x-1)\) --> \(x^2-3x+2=0\) --> \(x=1\) or \(x=2\).

Therefore, the product of the roots is 1*2=2.

Answer: C.

is there any alternate method to solve this problem..???

gmatclubot

Re: What is the product of all the solutions of x^2 - 4x + 6=3
[#permalink]
10 Oct 2016, 08:47

There’s something in Pacific North West that you cannot find anywhere else. The atmosphere and scenic nature are next to none, with mountains on one side and ocean on...

This month I got selected by Stanford GSB to be included in “Best & Brightest, Class of 2017” by Poets & Quants. Besides feeling honored for being part of...

Joe Navarro is an ex FBI agent who was a founding member of the FBI’s Behavioural Analysis Program. He was a body language expert who he used his ability to successfully...