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# Is root(x-5)^2 = 5 - x ?

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Manager
Joined: 01 Apr 2015
Posts: 57
Re: Is root(x-5)^2 = 5 - x ? [#permalink]

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09 Oct 2015, 07:40
Hi Bunuel,

Could you please explain this step |x−5|=5−x? --> is x−5≤0? --> is x≤5?. I know for all values less than or equal to 5 holds good for this equation, however, how do we arrive to that final equation using |x−5|=5−x.

What i do understand is that absolute value of anything has two values. When i simplify the equation above i get x=5 (when i take positive value) and x-5=x-5 (when i take negative value).

Thanks.
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Re: Is root(x-5)^2 = 5 - x ? [#permalink]

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09 Oct 2015, 07:51
Swaroopdev wrote:
Hi Bunuel,

Could you please explain this step |x−5|=5−x? --> is x−5≤0? --> is x≤5?. I know for all values less than or equal to 5 holds good for this equation, however, how do we arrive to that final equation using |x−5|=5−x.

What i do understand is that absolute value of anything has two values. When i simplify the equation above i get x=5 (when i take positive value) and x-5=x-5 (when i take negative value).

Thanks.

You know that $$\sqrt{x^2} = |x|$$ ---> $$\sqrt{x^2} = \pm x$$. Thus $$\sqrt{(x-5)^2} = |x-5|$$ --->$$\sqrt{(x-5)^2} = \pm (x-5)$$....(1)

Additionally, you know that $$\sqrt {x} \geq 0$$ (for GMAT purposes!)....(2)

Thus based on (1) and (2) above,

Coming back to your question, the questions asks whether $$\sqrt{(x-5)^2} = 5-x$$. This can only be possible when $$x\leq 5$$ as if x>5 then 5-x will become <0 and will go against (2) above. The most x can go is = 5....(3)

Also, from (1), $$\sqrt{x^2} = - x$$ only when x<0 ...(4)

Thus from (3) and (4), you get the rephrase of the question asked as is $$x \leq 5$$ ?

Hope this helps.
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Re: Is root(x-5)^2 = 5 - x ? [#permalink]

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10 Feb 2017, 11:06
SQRT( (x-5)^2 =5-x?

SQRT ( (x-5)^2 = |x-5|

|x-5| = 5-x

Now the above equation has two values

x-5 = 5-x or -(x-5) = 5-x depending on whether (x-5) > or < 0

1. -x|x| > 0

the above is possible only for x < 0, therefore (x-5) < 0

|x-5| = 5-x this becomes - - - > -(x-5) = 5-x and hence sufficient

2. Clearly states that (x-5) < 0 so this is sufficient and the answer is D

General takeaway here:
"squaring a quantity, and then square-rooting, is equivalent to taking the absolute value."
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Anaira Mitch

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Re: Is root(x-5)^2 = 5 - x ? [#permalink]

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22 Mar 2018, 08:54
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Re: Is root(x-5)^2 = 5 - x ?   [#permalink] 22 Mar 2018, 08:54

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