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# What is the value of |x|

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Manager
Joined: 10 Jan 2010
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What is the value of |x|  [#permalink]

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12 Apr 2012, 21:55
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What is the value of |x|

(1) |x^2 + 16| – 5 = 27

(2) x^2 = 8x – 16

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Joined: 10 Jan 2010
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12 Apr 2012, 22:00
pls clarify my doubt here.

statement 2 gives me the value x=4. Hence, MGMAT expln says |x| = 4 and sufficient to answer the question.

If i think inequalities as ranges in the number line, then statement 2 occupies one value x =4.

However, |x| = 4 has two values and occupies two positions +4 , -4 on the number line.
Hence, i did not choose statement 2.

Pls tell me where i went wrong in answering the q?.
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12 Apr 2012, 23:01
1
maheshsrini wrote:
pls clarify my doubt here.

statement 2 gives me the value x=4. Hence, MGMAT expln says |x| = 4 and sufficient to answer the question.

If i think inequalities as ranges in the number line, then statement 2 occupies one value x =4.

However, |x| = 4 has two values and occupies two positions +4 , -4 on the number line.
Hence, i did not choose statement 2.

Pls tell me where i went wrong in answering the q?.

First of all there are no inequalities in the question.

Next, let me ask you a question: if x=4 then what does |x| equal to? |4|=4. If it were |x|=4 and we were asked to find the value of x, then yes there would be two solutions x=4 or x=-4. So, it should be the other way around.

Complete solution.

What is the value of |x|

Notice that we are asked to find the absolute value of x (|x|).

(1) |x^2 + 16| – 5 = 27 -->|x^2+16|=32 --> x^2=16 --> |x|=4. Sufficient.

(2) x^2 = 8x – 16 --> x^2-8x+16=0 --> (x-4)^2=0 --> x=4 --> |x|=4. Sufficient.

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27 Jun 2012, 11:48
Bunuel wrote:
maheshsrini wrote:
pls clarify my doubt here.

statement 2 gives me the value x=4. Hence, MGMAT expln says |x| = 4 and sufficient to answer the question.

If i think inequalities as ranges in the number line, then statement 2 occupies one value x =4.

However, |x| = 4 has two values and occupies two positions +4 , -4 on the number line.
Hence, i did not choose statement 2.

Pls tell me where i went wrong in answering the q?.

First of all there are no inequalities in the question.

Next, let me ask you a question: if x=4 then what does |x| equal to? |4|=4. If it were |x|=4 and we were asked to find the value of x, then yes there would be two solutions x=4 or x=-4. So, it should be the other way around.

Complete solution.

What is the value of |x|

Notice that we are asked to find the absolute value of x (|x|).

(1) |x^2 + 16| – 5 = 27 -->|x^2+16|=32 --> x^2=16 --> |x|=4. Sufficient.

(2) x^2 = 8x – 16 --> x^2-8x+16=0 --> (x-4)^2=0 --> x=4 --> |x|=4. Sufficient.

Question! I can see how you (2) is sufficient. But I went about (1) in a different manner--

|x^2+16|-5=27
|x^2+16|=32
Then I broke it out into two parts...
a) x^2+16=32
x^2-16=0
(x-4)(x+4)=0
|x|=4

b) x^2+16=-32
x^2=-48... and then I kind of got stuck there.

Can you tell me where I went wrong with this logic?
Thank you!!!
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27 Jun 2012, 11:55
2
chris558 wrote:
Bunuel wrote:
maheshsrini wrote:
pls clarify my doubt here.

statement 2 gives me the value x=4. Hence, MGMAT expln says |x| = 4 and sufficient to answer the question.

If i think inequalities as ranges in the number line, then statement 2 occupies one value x =4.

However, |x| = 4 has two values and occupies two positions +4 , -4 on the number line.
Hence, i did not choose statement 2.

Pls tell me where i went wrong in answering the q?.

First of all there are no inequalities in the question.

Next, let me ask you a question: if x=4 then what does |x| equal to? |4|=4. If it were |x|=4 and we were asked to find the value of x, then yes there would be two solutions x=4 or x=-4. So, it should be the other way around.

Complete solution.

What is the value of |x|

Notice that we are asked to find the absolute value of x (|x|).

(1) |x^2 + 16| – 5 = 27 -->|x^2+16|=32 --> x^2=16 --> |x|=4. Sufficient.

(2) x^2 = 8x – 16 --> x^2-8x+16=0 --> (x-4)^2=0 --> x=4 --> |x|=4. Sufficient.

Question! I can see how you (2) is sufficient. But I went about (1) in a different manner--

|x^2+16|-5=27
|x^2+16|=32
Then I broke it out into two parts...
a) x^2+16=32
x^2-16=0
(x-4)(x+4)=0
|x|=4

b) x^2+16=-32
x^2=-48... and then I kind of got stuck there.

Can you tell me where I went wrong with this logic?
Thank you!!!

$$x^2+16=non-negative+positive=positive$$, so this expression cannot equal to negative number (-16). Next, since $$x^2+16>0$$ then $$|x^2+16|=x^2+16$$ and from $$|x^2 + 16| = 32$$ we can directly write $$x^2+16=32$$ --> $$x^2=32$$ --> $$|x|=4$$.

Hope it's clear.
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Re: What is the value of |x|  [#permalink]

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27 Jun 2012, 13:39

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Re: What is the value of |x|  [#permalink]

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28 Jun 2012, 01:12
1
maheshsrini wrote:
What is the value of |x|

(1) |x^2 + 16| – 5 = 27

(2) x^2 = 8x – 16

Statement (1):

|x^2 + 16| = 32 can be broken down into two:

(1) x^2 + 16 = 32 --> x^2 = 16 --> x = +4 and -4 --> So, the value of |X| is 4

(2) x^2 + 16 = -32 --> x^2 = -48 <-- this can't be the case. So this possibility doesn't hold.

Statement (1) is sufficient.

Statement (2):

x^2 = 8x - 16
(x-4)(x-4) = 0

x = 4

Statement (2) is sufficient.

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What is the value of |x|  [#permalink]

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29 May 2015, 01:46
1
maheshsrini wrote:
What is the value of |x|

(1) |x^2 + 16| – 5 = 27

(2) x^2 = 8x – 16

Love seeing the modes and the inequalities surrounding it , i often used to be scared until one day i realised what exactly is a inequality
Something that is not equal and a new world was born but the problem was when you mix absolute values in it and that is scary ,lol

Coming to the question, I would like to share a shortcut
Whenever you have a | x + a| = c , go ahead with x+a =+/- c and solve for each possibility of positive and negative
This way , you will save yourself lot of time
But then again revisiting basics , we know that square of a number can never be negative and lets see how that evolves out in this question prompt

(1) |x^2 + 16| – 5 = 27

can be written as |x^2 + 16|= 32

(a) Positive value

using our trick for positive value x^2 + 16 = 32
So , x^2 = 16
So x^2 - 16 = 0
Voila isn't the sweet boy looking familiar ...

Let me share another concept with you ...this is a equation with degree 2 , so it will have 2 factors
x^2 = 16 doesn't mean x=4 ...it mean there are two identical factors each of which is 4
Likewise whenever you have a equation having 3 degrees , always know that it shall have 3 factors

But anyways , if both value of x is 4 , isn't |x|=4 ....
easy to understand always know that |number| is distance of that number from zero on number line ...
So quite obviously |4|=4 ..

But wait , what about our second choice

(a) Negative value

There are two reasons it wont work
firstly , x^2 is positive and secondly 16 is positive as well
So when you add two positive numbers , GMAT will punish you if you show RHS to be negative

But say you do x^2 + 16 = -32
X^2=-48
Can square of a number be positive 1^2=1 , -1^2=1 ..
In fact square and mode are those sweet little devils which hide the sign and therefore when you have them , you actually dunno what the number is
But anyways , if you have square of a number , rest assured that it's cant be negative

And so my friend , we have to reject this solution

And so pick up the positive solution and therein lies our sufficiency for this part

So , hmmmm...lets see A is alive because statement 1 is sufficient , we can bid adieu to C,E( but they are still nice fellows ..maybe we need them later but not for this question , sowwie !)

(2) Statement 2

-x^2 = 8x – 16

Rearrange it a bit and it boils down to again (x-4)^2 = 0 again two roots and each of them 4
So , mode of 4 is 4 and sweet little boy D is our answer here
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Re: What is the value of |x|  [#permalink]

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06 Sep 2017, 05:28
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