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how do you solve absolute value equations using algebra?

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how do you solve absolute value equations using algebra?  [#permalink]

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New post 25 Feb 2018, 18:14
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I'm a bit confused on how to use algebra to solve absolute value equations.

For example: If |x^2 − 12| = x, which of the following could be the value of x?

A. –4
B. –3
C. 1
D. 2
E. 3

I solved this easily using substitution, but I want to know how to solve it using algebra. I saw a similar problem and tried to copy over the steps but apparently it doesn't work algebraically here. Not sure what I am missing.

From my understanding you assume what's in the absolute value to be positive and negative, then solve for x. Assuming If |x^2 − 12| is positive, we get
x^2 - 12 = +x
x^2 -x -12 = 0
(x-4)(x+3)
x=4 or -3
Then do we just eliminate x=-3 since x has to be positive?

Next, assuming x is negative
x^2 - 12 = -x
x^2 + x - 12 = 0
(x+4)(x-3)
x=-4 or +3
Here, do we just eliminate x=+3 since we assumed x has to be negative?
Or should we not have done this step all together since we cannot assume x to be negative since |x^2 − 12| will never be a negative value?
In which case, how do we get the correct answer of E. x = 3.

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Re: how do you solve absolute value equations using algebra?  [#permalink]

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New post 25 Feb 2018, 18:33
jasonfodor wrote:
I'm a bit confused on how to use algebra to solve absolute value equations.

For example: If |x^2 − 12| = x, which of the following could be the value of x?

A. –4
B. –3
C. 1
D. 2
E. 3

I solved this easily using substitution, but I want to know how to solve it using algebra. I saw a similar problem and tried to copy over the steps but apparently it doesn't work algebraically here. Not sure what I am missing.

From my understanding you assume what's in the absolute value to be positive and negative, then solve for x. Assuming If |x^2 − 12| is positive, we get
x^2 - 12 = +x
x^2 -x -12 = 0
(x-4)(x+3)
x=4 or -3
Then do we just eliminate x=-3 since x has to be positive?

Next, assuming x is negative
x^2 - 12 = -x
x^2 + x - 12 = 0
(x+4)(x-3)
x=-4 or +3
Here, do we just eliminate x=+3 since we assumed x has to be negative?
Or should we not have done this step all together since we cannot assume x to be negative since |x^2 − 12| will never be a negative value?
In which case, how do we get the correct answer of E. x = 3.


Firstly x is equal to modulus so x is positive..
Now you have to take modulus as POSITIVE or NEGATIVE
1) \(|x^2-12|=x.....x^2-12=x.....x^2-x-12=0....(x-4)(x+3)=0\)...
x=4 or -3
2)\(|x^2-12|=x.....-(x^2-12)=x.....x^2+x-12=0....(x+4)(x-3)=0\)...
x=-4 or 3

So two values come out 4 and 3..
4 is not given in Choices but 3 is given
So answer 3

Hope it helps
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


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Re: how do you solve absolute value equations using algebra?  [#permalink]

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New post 25 Feb 2018, 20:15
jasonfodor wrote:
I'm a bit confused on how to use algebra to solve absolute value equations.

For example: If |x^2 − 12| = x, which of the following could be the value of x?

A. –4
B. –3
C. 1
D. 2
E. 3

I solved this easily using substitution, but I want to know how to solve it using algebra. I saw a similar problem and tried to copy over the steps but apparently it doesn't work algebraically here. Not sure what I am missing.

From my understanding you assume what's in the absolute value to be positive and negative, then solve for x. Assuming If |x^2 − 12| is positive, we get
x^2 - 12 = +x
x^2 -x -12 = 0
(x-4)(x+3)
x=4 or -3
Then do we just eliminate x=-3 since x has to be positive?

Next, assuming x is negative
x^2 - 12 = -x
x^2 + x - 12 = 0
(x+4)(x-3)
x=-4 or +3
Here, do we just eliminate x=+3 since we assumed x has to be negative?
Or should we not have done this step all together since we cannot assume x to be negative since |x^2 − 12| will never be a negative value?
In which case, how do we get the correct answer of E. x = 3.


Discussed here: if-x-2-12-x-which-of-the-following-could-be-the-value-of-x-217526.html In case of any questions please post in that topic. Thank you.

--== Message from the GMAT Club Team ==--

THERE IS LIKELY A BETTER DISCUSSION OF THIS EXACT QUESTION.
This discussion does not meet community quality standards. It has been retired.


If you would like to discuss this question please re-post it in the respective forum. Thank you!

To review the GMAT Club's Forums Posting Guidelines, please follow these links: Quantitative | Verbal Please note - we may remove posts that do not follow our posting guidelines. Thank you.

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Re: how do you solve absolute value equations using algebra? &nbs [#permalink] 25 Feb 2018, 20:15
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