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# Is |x|<1?

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Manager
Joined: 21 Jun 2010
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28 Jul 2010, 03:56
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Is |x| < 1 ?

(1) |x + 1| = 2|x – 1|

(2) |x – 3| > 0
[Reveal] Spoiler: OA

Last edited by Bunuel on 04 Dec 2012, 03:03, edited 1 time in total.
Renamed the topic and edited the question.
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Re: Inequality - absolute on both sides [#permalink]

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28 Jul 2010, 04:22
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mn2010 wrote:
The question is

Is |x| < 1 ?

(1) |x + 1| = 2|x – 1|

(2) |x – 3| > 0

I always get confused how to approach the equality or inequality with absolute signs on both sides (statement 1). Any1 knows an efficient way to approach this ?

Is $$|x| < 1$$?

Is $$|x| < 1$$, means is $$x$$ in the range (-1,1) or is $$-1<x<1$$ true?

(1) $$|x + 1| = 2|x - 1|$$
Two key points: $$x=-1$$ and $$x=1$$ (key points are the values of $$x$$ when absolute values equal to zero), thus three ranges to check:
---------{-1}--------{1}---------

A. $$x<-1$$ (blue range) --> $$|x + 1| = 2|x - 1|$$ becomes: $$-x-1=2(-x+1)$$ --> $$x=3$$, not OK, as this value is not in the range we are checking ($$x<-1$$);
B. $$-1\leq{x}\leq{1}$$ (green range) --> $$|x + 1| = 2|x - 1|$$ becomes: $$x+1=2(-x+1)$$ --> $$x=\frac{1}{3}$$. OK, as this value is in the range we are checking ($$-1\leq{x}\leq{1}$$);
C. $$x>1$$ (red range) --> $$|x + 1| = 2|x - 1|$$ becomes: $$x+1=2(x-1)$$ --> $$x=3$$. OK, as this value is in the range we are checking ($$x>1$$).

So we got TWO values of $$x$$ (two solutions): $$\frac{1}{3}$$ and $$3$$, first is in the range (-1,1) but second is out of the range. Not sufficient.

Alternately we could just expand absolute values of RHS and LHS with same sign (for example both LHS and RHS positive: $$x + 1 = 2(x - 1)$$) and then with different sign (for example LHS positive and RHS negative $$x + 1 = 2(-x + 1)$$), solve for $$x$$ both equations, and finally check whether the solutions satisfy $$|x + 1| = 2|x - 1|$$.

OR: we can square given equation to get rid of the modulus: $$(x + 1)^2 = 4(x - 1)^2$$ --> $$3x^2-10x+3=0$$ --> $$x=3$$ or $$x=\frac{1}{3}$$.

(2) $$|x - 3|>{0}$$. Absolute value is always non-negative, more than or equal to zero: $$|some \ expression|\geq{0}$$. We are told that absolute value of $$x-3$$ is MORE than zero, so just it says that $$|x-3|\neq{0}$$, which simply means that $$x\neq{3}$$. But we don't know whether $$x$$ is in the range (-1,1) or not.

(1)+(2) $$x=\frac{1}{3}$$ or $$x=3$$ AND $$x\neq{3}$$ --> means $$x$$ can have only value $$\frac{1}{3}$$, which is in the range (-1,1). Sufficient.

Hope it helps.
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Re: need help on absolute values [#permalink]

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03 Feb 2011, 18:53
Merging similar topics.

For more on absolute values check Walker's post: math-absolute-value-modulus-86462.html

For practice check collection of 13 tough inequalities and absolute values questions with detailed solutions: inequality-and-absolute-value-questions-from-my-collection-86939.html

Also: some-inequalities-questions-93760.html

More DS questions on absolute value: search.php?search_id=tag&tag_id=37

More PS questions on absolute value: search.php?search_id=tag&tag_id=58

Hope it helps.
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Re: Inequality - absolute on both sides [#permalink]

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04 Feb 2011, 10:58
[quote="Bunuel"][quote="mn2010"]
B. $$-1\leq{x}\leq{1}$$ (green range) --> $$|x + 1| = 2|x - 1|$$ becomes: $$x+1=2(-x+1)$$ --> $$x=\frac{1}{3}$$.

Hi Bunuel:

Can you please explain why, in B above, we must NOT change the sign in |x+1| leaving it as x+1, and must change the sign in 2|x-1| making it 2(-x+1)?

Thank you.
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Re: Inequality - absolute on both sides [#permalink]

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04 Feb 2011, 11:13
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Matt1177 wrote:
Bunuel wrote:
mn2010 wrote:

B. $$-1\leq{x}\leq{1}$$ (green range) --> $$|x + 1| = 2|x - 1|$$ becomes: $$x+1=2(-x+1)$$ --> $$x=\frac{1}{3}$$.

Hi Bunuel:

Can you please explain why, in B above, we must NOT change the sign in |x+1| leaving it as x+1, and must change the sign in 2|x-1| making it 2(-x+1)?

Thank you.

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 B. if $$-1\leq{x}\leq{1}$$ (green range) --> then $$x+1\geq{0}$$ (try some value of $$x$$ from the given range to check: for example $$x=0$$) so $$|x+1|=x+1$$ BUT $$x-1\leq{0}$$ ] (again try some value of $$x$$ from the given range to check: for example $$x=0$$) so $$|x-1|=-(x-1)$$ thus $$|x + 1| = 2|x - 1|$$ becomes: $$x+1=2(-x+1)$$.

For more check: math-absolute-value-modulus-86462.html

Hope it's clear.
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Re: Inequality - absolute on both sides [#permalink]

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04 Feb 2011, 12:19
2
KUDOS
The question is

Is |x| < 1 ?

Asking whether x falls between 1 and -1, exclusive.

if |x|<1
x<1; 0.5, 0.2,0
and
x>-1; -0.5,-0.2

or -1<x<1

Moment you write x=1 or x=-1; the |x|<1 becomes false.

(1) |x + 1| = 2|x – 1|

For modulus on both sides:

Case I.
solve the equation with no sign change

so +(x+1) = +2(x-1)
x+1=2x-2
x=3. Not between -1 and 1.

Case II.
solve the equation with sign change on one side. Either RHS or LHS. Let's do the sign change on LHS

so -(x+1) = +2(x-1)
-x-1=2x-2
3x=1.
x=1/3
x is between -1 and 1.

CaseII(b):
Even if we did sign change on RHS, we would have gotten the same result. Let's try
+(x+1) = -2(x-1)
x+1=-2x+2
3x=1
x=1/3. Same result as before.

So; sign change should be done for either LHS or RHS.

Now, we have two solutions for x; 1/3(between -1 and 1), 3(not between -1 and 1)

Not sufficient.

###Also please substitute these factors of x into the main equation and check whether the factors indeed satisfy the equation. because say if 3 didn't satisfy the equation and 1/3 does. The statement would be sufficient.###

(2) |x – 3| > 0

Here modulus only on LHS;
So, try this with both signs;

+(x-3) > 0
x-3>0
x>3

and

-(x-3) > 0
-x+3>0
-x>-3
x<3

Here x can be either less than 3 or more than 3. Not 3.

However; this doesn't tell us definitively whether x lies between -1 and 1.

Not Sufficient.

Using both:

We know x can be either (1/3 or 3) by 1st statement. Second statement tells us that x can not be 3.

The only value thus left is 1/3.

Sufficient.

Ans: "C"
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Re: Inequality - absolute on both sides [#permalink]

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04 Feb 2011, 14:47
Matt1177 wrote:
Bunuel wrote:
For B. if $$-1\leq{x}\leq{1}$$ (green range) --> then $$x+1\geq{0}$$ (try some value of $$x$$ from the given range to check: for example $$x=0$$) so $$|x+1|=x+1$$ BUT $$x-1\leq{0}$$ ] (again try some value of $$x$$ from the given range to check: for example $$x=0$$) so $$|x-1|=-(x-1)$$ thus $$|x + 1| = 2|x - 1|$$ becomes: $$x+1=2(-x+1)$$.

So, as I understand, we need to plug the value from the available range into the modulus. If the expression in the modulus becomes negative of zero, we must change the sign. And if the expression becomes positive or zero, then we should leave it as it is. Am I correct?

Thanks again for your help, Bunuel.

Yes. You should really try the links I've provided in the above posts.

Matt1177 wrote:
fluke wrote:
Case II.
solve the equation with sign change on one side. Either RHS or LHS. Let's do the sign change on LHS

So, we cannot change the signs on both LHS and RHS at once?

No that's not the point. When you have absolute values on both sides of the equation expansion can be either + + (or which is the same - - because $$x + 1 = 2(x - 1)$$ is the same as $$-(x + 1) = -2(x - 1)$$) OR + - (or which is the same - + because $$x + 1 = -2(x - 1)$$ is the same as $$-(x + 1)=2(x - 1)$$) so basically only two options.

So as I've written in my solution: "alternately you could just expand absolute values of RHS and LHS with same sign (for example both LHS and RHS positive: $$x + 1 = 2(x - 1)$$) and then with different sign (for example LHS positive and RHS negative $$x + 1 = 2(-x + 1)$$), solve for $$x$$ both equations, and finally check whether the solutions satisfy $$|x + 1| = 2|x - 1|$$."

Here are two more links which might helps to understand this:
absolute-value-ds-100357.html
some-inequalities-questions-93760.html

Hope it helps.
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Re: Inequality - absolute on both sides [#permalink]

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05 Feb 2011, 08:41
Bunuel...if the second statement was (2) |x – 3| = 0 then in that case could we say

x= 3.
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Re: Inequality - absolute on both sides [#permalink]

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05 Feb 2011, 08:51
ajit257 wrote:
Bunuel...if the second statement was (2) |x – 3| = 0 then in that case could we say

x= 3.

If (2) were |x – 3| = 0 then yes, we would have that x=3 and in this case this statement would be sufficient, as we could answer NO to the question whether -1<x<1.
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Re: Inequalities, Is |X| < 1 ? [#permalink]

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20 Jul 2011, 02:15
scbguy wrote:
I see the answer as A, obviously I'm wrong but I don't see how x is 1/3 in statement 1

Posted from GMAT ToolKit

|x+1| = 2|x-1|
(x+1) = 2*-(x-1)
x+1 = -2x + 2
3x = 1
x = 1/3

|1/3+1| = 4/3
2|1/3-3/3| = 2*2/3 = 4/3

1/3 is a solution to 1)
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Re: Inequalities, Is |X| < 1 ? [#permalink]

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20 Jul 2011, 02:17
scbguy wrote:
I see the answer as A, obviously I'm wrong but I don't see how x is 1/3 in statement 1

Posted from GMAT ToolKit

(1) |x + 1| = 2|x – 1|

This has 2 cases.. X>0 and X<0
If X>0 , then X+1 = 2(x-1)
If X<0 , then X+1 = -2(x-1)

Solving these equations we get X= 3 or X= 1/3. Since we have YES and NO situation => Not sufficient

(2) |x – 3| > 0

Solving this equation , we get x>3 or X<3, in either cases, X<> 3. So not sufficient.

(1) + (2) ==> X= 1/3 . Since X<> 3.

Regards,
Mustu
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Re: Inequalities, Is |X| < 1 ? [#permalink]

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20 Jul 2011, 05:55
Mustu, for (1) |x + 1| = 2|x – 1| when looking at X<0 case, how come you multiply only right-hand side by negative, but not both sides? thanks
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Re: Inequalities, Is |X| < 1 ? [#permalink]

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21 Jul 2011, 00:02
tt2011 wrote:
Mustu, for (1) |x + 1| = 2|x – 1| when looking at X<0 case, how come you multiply only right-hand side by negative, but not both sides? thanks

We have to consider 4 cases totally..

1) + +
2) + -
3) - +
4) - -

If u analyse more closely, then u will find that case 1 and 4 ( - and - cancels out) are the same and cases 2 and 3 are the same( + and - 0r - and + are the same). Hope this is clear.

Regards,
Mustu
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Re: Inequalities, Is |X| < 1 ? [#permalink]

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22 Jul 2011, 14:34
So does (2) implies that x IS NOT = 3? Hence we can use x =1/3 as our value?
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Re: Inequalities, Is |X| < 1 ? [#permalink]

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22 Jul 2011, 18:48
kanishk wrote:
So does (2) implies that x IS NOT = 3? Hence we can use x =1/3 as our value?

Exactly. So 1) gives us two options, but we don't know which x is. 2) tells us that x does not equal 3, so we know that x = 1/3
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Re: Inequalities, Is |X| < 1 ? [#permalink]

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24 Jul 2011, 13:28
kanishk wrote:
So does (2) implies that x IS NOT = 3? Hence we can use x =1/3 as our value?

Yes, you got it right
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Re: Inequalities, Is |X| < 1 ? [#permalink]

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31 Jul 2011, 10:19
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I think its quicker just to square both sides. 30 sec you have the answer and little room for mistakes.
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Re: Inequalities, Is |X| < 1 ? [#permalink]

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31 Jul 2011, 20:33
Thanks 144144

This is a wonderful method

statement 1 :
(X+1)^2 = 4 (x-1)^2

this translated to a quadratic equation

X^2 +6X - 1 = 0 clearly there are 2 answers so not sufficient

Statement 2 translates to X not eqaul to 3

combining both, we have our solution

Kudos to you
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Re: Inequalities, Is |X| < 1 ? [#permalink]

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06 Sep 2011, 03:11
Quote:
Is |x| < 1 ?

(1) |x + 1| = 2|x – 1|

(2) |x – 3| > 0

From Statement 1
square both sides:
--> 2^2+2x+1=4x^2-8x+4
--> 0 = 3x^2-10x+3
--> 0=(3x-1)(x-3)
x=3 or x=1/3
Hence, insufficient.

From Statement 2
values of x: any number larger or smaller than 3 (i.e. x not equals to 3)
Hence, insufficient.

From statement 1+2

Sufficient.

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Re: Inequalities, Is |X| < 1 ? [#permalink]

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14 Sep 2011, 05:35
mustu wrote:
tt2011 wrote:
Mustu, for (1) |x + 1| = 2|x – 1| when looking at X<0 case, how come you multiply only right-hand side by negative, but not both sides? thanks

We have to consider 4 cases totally..

1) + +
2) + -
3) - +
4) - -

If u analyse more closely, then u will find that case 1 and 4 ( - and - cancels out) are the same and cases 2 and 3 are the same( + and - 0r - and + are the same). Hope this is clear.

Regards,
Mustu

Could you post all the case in this question and reason why are these cases ?I have a lower hand in Inequalities ;Therefore , I can't figure out what are referring about .
Re: Inequalities, Is |X| < 1 ?   [#permalink] 14 Sep 2011, 05:35

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