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605-655 Level|   Absolute Values|   Inequalities|         
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Is the square root of (x - 5)^2 = 5 - x ? (1) -x|x| > 0 (2) 5 - x > 0 13 Jan 2007, 11:52
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Is \(\sqrt{(x-5)^2} = 5 - x\)?


(1) \(-x|x| > 0\)

(2) \(5 - x > 0\)
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Is the square root of (x - 5)^2 = 5 - x ? (1) -x|x| > 0 (2) 5 - x > 0 09 Sep 2009, 15:41
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dhushan
Just want to make sure that my line of thinking is right. I got D as well.

sqrt(x-5)^2 = 5-x
==> |x-5| = 5-x

which means that if

x-5<0, -(x-5) = 5-x = no solution, therefore x<5

x-5>0, x-5 = 5-x, which means that x>5 and x = 5

from (1) -x|x| > 0, we know that x has to be negative -(-x)|-x| is the only way to get a number greater than 0. Therefore, this means that of the three possible solutions for x, only this x<5 hold true.

from (2) 5-x>0, therefore x<5.

Can someone please point out if there is something wrong with my reasoning.

Is \(\sqrt{(x-5)^2}=5-x\)?

First of all, recall that \(\sqrt{x^2}=|x|\).

Is \(\sqrt{(x-5)^2}=5-x\)? --> is \(|x-5|=5-x\)? --> is \(x-5\leq{0}\)? --> is \(x\leq{5}\)?

(1) \(-x|x| > 0\) --> \(|x|\) is never negative (positive or zero), so for \(-x|x|\) to be positive, \(-x\) must be positive \(-x>0\) --> \(x<0\). Sufficient.

(2) \(5-x>0\) --> \(x<5\). Sufficient.

Answer: D.
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Is the square root of (x - 5)^2 = 5 - x ? (1) -x|x| > 0 (2) 5 - x > 0 06 Sep 2010, 12:30
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metallicafan
Is \(sqrt(x-5)^2 = 5-x?\)

(1) -xlxl > 0
(2) 5-x > 0

PS. Is always this true?: \(sqrt{x^2}\) = lxl ?

Is \(\sqrt{(x-5)^2}=5-x\)?

Remember: \(\sqrt{x^2}=|x|\).

So "is \(\sqrt{(x-5)^2}=5-x\)?" becomes: is \(|x-5|=5-x\)?

\(|x-5|=5-x\) is true only for \(x\leq{5}\), as in this case \(\{LHS=|x-5|=5-x\}=\{RHS=5-x\}\). So we have that if \(x\leq{5}\), then \(|x-5|=5-x\) is true.

Basically question asks is \(x\leq{5}\)?

(1) \(-x|x| > 0\) --> \(|x|\) is never negative (positive or zero), so in order to have \(-x|x| > 0\), \(-x\) must be positive \(-x>0\) --> \(x<0\), so \(x\) is less than 5 too. Sufficient.

(2) \(5-x>0\) --> \(x<5\). Sufficient.

Answer: D.

Hope it helps.

P.S. Explanation of: \(\sqrt{x^2}=|x|\).

The point here is that as square root function cannot give negative result then \(\sqrt{some \ expression}\geq{0}\).

So \(\sqrt{x^2}\geq{0}\). But what does \(\sqrt{x^2}\) equal to?

Let's consider following examples:
If \(x=5\) --> \(\sqrt{x^2}=\sqrt{25}=5=x=positive\);
If \(x=-5\) --> \(\sqrt{x^2}=\sqrt{25}=5=-x=positive\).

So we got that:
\(\sqrt{x^2}=x\), if \(x\geq{0}\);
\(\sqrt{x^2}=-x\), if \(x<0\).

What function does exactly the same thing? The absolute value function: \(|x|=x\), if \(x\geq{0}\) and \(|x|=-x\), if \(x<0\). That is why \(\sqrt{x^2}=|x|\).
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Is the square root of (x - 5)^2 = 5 - x ? (1) -x|x| > 0 (2) 5 - x > 0 13 Jan 2007, 12:11
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For me (D) :)

sqrt( (x-5)^2) = 5 - x ?
<=> |x-5| = 5-x ?
<=> |5-x| = 5-x ?

This is true if 5-x >= 0 <=> x =< 5

So we end up with checking if x =< 5 ?

Stat 1
-x |x| > 0
<=> x < 0 < 5

SUFF.

Stat 2
5 - x > 0
<=> x < 5

SUFF.
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Is the square root of (x - 5)^2 = 5 - x ? (1) -x|x| > 0 (2) 5 - x > 0 09 Sep 2009, 12:01
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Just want to make sure that my line of thinking is right. I got D as well.

sqrt(x-5)^2 = 5-x
==> |x-5| = 5-x

which means that if

x-5<0, -(x-5) = 5-x = no solution, therefore x<5

x-5>0, x-5 = 5-x, which means that x>5 and x = 5

from (1) -x|x| > 0, we know that x has to be negative -(-x)|-x| is the only way to get a number greater than 0. Therefore, this means that of the three possible solutions for x, only this x<5 hold true.

from (2) 5-x>0, therefore x<5.

Can someone please point out if there is something wrong with my reasoning.
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Is the square root of (x - 5)^2 = 5 - x ? (1) -x|x| > 0 (2) 5 - x > 0 31 Dec 2009, 09:40
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The question basically wants to know if x<=5 else RHS will be x-5

Statement 1
-x|x|>0
or
x|x|<0 (multiply by -1 both sides and reverse the sign)
either x<0 or |x|<0
since |x| is always positive or 0 x<0 is true.
if x<0 then x<5 hence sufficient

Statement 2
5-x>0
5>x
This is what we are looking for hence sufficient

Answer is D.
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metallicafan
Is \(sqrt(x-5)^2 = 5-x?\)

(1) -xlxl > 0
(2) 5-x > 0

PS. Is always this true?: \(sqrt{x^2}\) = lxl ?

Is \(\sqrt{(x-5)^2}=5-x\)?

Remember: \(\sqrt{x^2}=|x|\).

So "is \(\sqrt{(x-5)^2}=5-x\)?" becomes: is \(|x-5|=5-x\)?


\(|x-5|=5-x\) is true only for \(x\leq{5}\), as in this case \(\{LHS=|x-5|=5-x\}=\{RHS=5-x\}\). So we have that if \(x\leq{5}\), then \(|x-5|=5-x\) is true.

Basically question asks is \(x\leq{5}\)?

(1) \(-x|x| > 0\) --> \(|x|\) is never negative (positive or zero), so in order to have \(-x|x| > 0\), \(-x\) must be positive \(-x>0\) --> \(x<0\), so \(x\) is less than 5 too. Sufficient.

(2) \(5-x>0\) --> \(x<5\). Sufficient.

Answer: D.

Hope it helps.


Hi Bunuel,
I get confused in the following concept. Can you please help me:


\(-x|x| > 0\) --> \(|x|\) is never negative (positive or zero), so in order to have \(-x|x| > 0\), \(-x\) must be positive \(-x>0\) --> \(x<0\), so \(x\) is less than 5 too

Can't we solve it as follows:
As \(|x|\) is never negative -> \(-x|x| > 0\) = -x*x = -x^2 >0 = x^2<0 (multiplying by -ve sign and flipping sign)
x^2<0 =>
\(sqrt{x^2}\) <0
=> lxl <0 (as \(sqrt{x^2}\) =lxl )

Since lxl cannot be negative and lxl <0 that implies X<0

I have reached to same conclusion as yours but wanted to confirm if my approach is right. Please explain.Thanks
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Bunuel
metallicafan
Is \(sqrt(x-5)^2 = 5-x?\)

(1) -xlxl > 0
(2) 5-x > 0

PS. Is always this true?: \(sqrt{x^2}\) = lxl ?

Is \(\sqrt{(x-5)^2}=5-x\)?

Remember: \(\sqrt{x^2}=|x|\).

So "is \(\sqrt{(x-5)^2}=5-x\)?" becomes: is \(|x-5|=5-x\)?


\(|x-5|=5-x\) is true only for \(x\leq{5}\), as in this case \(\{LHS=|x-5|=5-x\}=\{RHS=5-x\}\). So we have that if \(x\leq{5}\), then \(|x-5|=5-x\) is true.

Basically question asks is \(x\leq{5}\)?

(1) \(-x|x| > 0\) --> \(|x|\) is never negative (positive or zero), so in order to have \(-x|x| > 0\), \(-x\) must be positive \(-x>0\) --> \(x<0\), so \(x\) is less than 5 too. Sufficient.

(2) \(5-x>0\) --> \(x<5\). Sufficient.

Answer: D.

Hope it helps.


Hi Bunuel,
I get confused in the following concept. Can you please help me:


\(-x|x| > 0\) --> \(|x|\) is never negative (positive or zero), so in order to have \(-x|x| > 0\), \(-x\) must be positive \(-x>0\) --> \(x<0\), so \(x\) is less than 5 too

Can't we solve it as follows:
As \(|x|\) is never negative -> \(-x|x| > 0\)= -x*x = -x^2 >0 = x^2<0 (multiplying by -ve sign and flipping sign)
x^2<0 =>
\(sqrt{x^2}\) <0
=> lxl <0 (as \(sqrt{x^2}\) =lxl )

Since lxl cannot be negative and lxl <0 that implies X<0

I have reached to same conclusion as yours but wanted to confirm if my approach is right. Please explain.Thanks

This approach is not right.

The red parts are not correct.

\(-x|x| > 0\) cannot be written as \(-x*x>0\), as \(|x|\geq{0}\) does not mean \(x\) itself cannot be negative --> \(|x|=x\) if \(x\geq{0}\) and \(|x|=-x\) if \(x\leq{0}\), so when you are writing \(x\) instead of \(|x|\) you are basically assuming that \(x\geq{0}\) and then in the end get the opposite result \(x<0\).

Next, \(x^2<0\) has no solution, square of a number cannot be negative, so no \(x\) can make this inequality hold true.
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Is the square root of (x - 5)^2 = 5 - x ? (1) -x|x| > 0 (2) 5 - x > 0 07 Sep 2010, 23:42
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I think for this question we dont need any statement... without statement itself it is possible to say if the equality is correct or wrong. Can someone comment on this... Bunuel please?
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Is the square root of (x - 5)^2 = 5 - x ? (1) -x|x| > 0 (2) 5 - x > 0 08 Sep 2010, 06:59
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amitjash
I think for this question we dont need any statement... without statement itself it is possible to say if the equality is correct or wrong. Can someone comment on this... Bunuel please?

This not correct. \(x\) must be less than or equal to 5 inequality \(\sqrt{(x-5)^2}=5-x\) to hold true. For example if \(x=10>5\) then \(\sqrt{(x-5)^2}=5\neq{-5}=5-x\)
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Is the square root of (x - 5)^2 = 5 - x ? (1) -x|x| > 0 (2) 5 - x > 0 19 Mar 2011, 20:20
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arjunrampal
Please explain the approach to solve the problem and point to any relevant material available in the GMAT club.

Attachment:
square_root_D.JPG

Show Spoiler
AO=D

Let me point out something here: You cannot square both sides to get
Is \((\sqrt{(x-5)^2})^2 = (5-x)^2\) ?

People sometimes get confused here. Why can you not square it?
It is a question similar to 'Is x = 5?' Can you square both sides here and change the question to 'Is \(x^2 = 25\)?' Please remember, they are not the same. x^2 can be 25 even if x is not 5 ( when x = -5, even then x^2 = 25).
Only if it is given to you that x = 5, then you can say that x^2 = 25.

You can rephrase the question in the following manner (and many more ways)

Is \((\sqrt{(x-5)^2}) = (5-x)\) ?
Is \(|x-5| = (5-x)\) ?
or Is \(|5-x| = (5-x)\)?
We know that |x| = x only when x >= 0
So \(|5-x| = (5-x)\) only when 5 - x >= 0 or when x <= 5

Stmnt 1: -x|x| > 0
Since |x| is always positive (or zero), -x must be positive too. So x must be negative.
If x < 0, then x is obviously less than 5. Sufficient.

Stmnt 2: 5 - x> 0
x < 5. Sufficient

Answer D
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Is the square root of (x - 5)^2 = 5 - x ? (1) -x|x| > 0 (2) 5 - x > 0 12 Jul 2015, 08:32
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kshitij89

Modulus always results in an positive value.
If x-5>0 then |x−5| should be equal to x-5
However, If x-5<0 then modulus would result the positive value of it i.e. -(x-5)=5-x

Thus, is |x−5|=5−x? --> is x−5<0? --> is x<5?

Solving as Bunuel ,you'll get D. :)
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Just want to make sure that my line of thinking is right. I got D as well.

sqrt(x-5)^2 = 5-x
==> |x-5| = 5-x

which means that if

x-5<0, -(x-5) = 5-x = no solution, therefore x<5

x-5>0, x-5 = 5-x, which means that x>5 and x = 5

from (1) -x|x| > 0, we know that x has to be negative -(-x)|-x| is the only way to get a number greater than 0. Therefore, this means that of the three possible solutions for x, only this x<5 hold true.

from (2) 5-x>0, therefore x<5.

Can someone please point out if there is something wrong with my reasoning.

Is \(\sqrt{(x-5)^2}=5-x\)?

First of all, recall that \(\sqrt{x^2}=|x|\).

Is \(\sqrt{(x-5)^2}=5-x\)? --> is \(|x-5|=5-x\)? --> is \(x-5\leq{0}\)? --> is \(x\leq{5}\)?

(1) \(-x|x| > 0\) --> \(|x|\) is never negative (positive or zero), so for \(-x|x|\) to be positive, \(-x\) must be positive \(-x>0\) --> \(x<0\). Sufficient.

(2) \(5-x>0\) --> \(x<5\). Sufficient.

Answer: D.
in the highlighted portion, why we can not also write that x is greater than or equal to 5. if we take out modulus, |x-5| can be positive and can be negative. there are two possible values of x. you consider only negative possibility. please help. i don't understand.
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Bunuel
dhushan
Just want to make sure that my line of thinking is right. I got D as well.

sqrt(x-5)^2 = 5-x
==> |x-5| = 5-x

which means that if

x-5<0, -(x-5) = 5-x = no solution, therefore x<5

x-5>0, x-5 = 5-x, which means that x>5 and x = 5

from (1) -x|x| > 0, we know that x has to be negative -(-x)|-x| is the only way to get a number greater than 0. Therefore, this means that of the three possible solutions for x, only this x<5 hold true.

from (2) 5-x>0, therefore x<5.

Can someone please point out if there is something wrong with my reasoning.

Is \(\sqrt{(x-5)^2}=5-x\)?

First of all, recall that \(\sqrt{x^2}=|x|\).

Is \(\sqrt{(x-5)^2}=5-x\)? --> is \(|x-5|=5-x\)? --> is \(x-5\leq{0}\)? --> is \(x\leq{5}\)?

(1) \(-x|x| > 0\) --> \(|x|\) is never negative (positive or zero), so for \(-x|x|\) to be positive, \(-x\) must be positive \(-x>0\) --> \(x<0\). Sufficient.

(2) \(5-x>0\) --> \(x<5\). Sufficient.

Answer: D.
in the highlighted portion, why we can not also write that x is greater than or equal to 5. if we take out modulus, |x-5| can be positive and can be negative. there are two possible values of x. you consider only negative possibility. please help. i don't understand.


Hello

I will try to explain. If x is >= 5 (greater than or equal to 5), then x-5 will be positive, and square root of (x-5)^2, will be equal to x-5 only, it will never be equal to 5-x.

Eg, lets take x=6, here x-5 = 1, which is positive. Here (x-5)^2 = 1^2 = 1, and its square root is 1, which is x-5 only, NOT 5-x.
But if we take x=4, here x-5 = -1, which is negative. Here (x-5)^2 = (-1)^2 = 1, and its square root is 1, which is NOT x-5: rather its 5-x.
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Hi Bunuel ,

Can you please tell me where I am doing wrong ?

I think for this question we don't need any statement because as the question is asking

Is sqrt(x-5)^2 = 5-x ?

We can write above one like this :

sqrt(x-5)^2 = -(x-5)

Now Squaring both sides

(x-5)^2 = {-(x-5) }^2

Hence , (x-5)^2 = (x-5)^2.

Please correct me, where I am getting it wrong.

Thanks
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Is the square root of (x - 5)^2 = 5 - x ? (1) -x|x| > 0 (2) 5 - x > 0 18 Aug 2019, 22:31
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a12bansal
Hi Bunuel ,

Can you please tell me where I am doing wrong ?

I think for this question we don't need any statement because as the question is asking

Is sqrt(x-5)^2 = 5-x ?

We can write above one like this :

sqrt(x-5)^2 = -(x-5)

Now Squaring both sides

(x-5)^2 = {-(x-5) }^2

Hence , (x-5)^2 = (x-5)^2.

Please correct me, where I am getting it wrong.

Thanks

I addressed this doubt here: https://gmatclub.com/forum/is-root-x-5- ... ml#p777283

Also, you can only square equations when both sides are non-negative. 5 - x COULD be negative when x > 5 so when squaring you might get wrong result.
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Is the square root of (x - 5)^2 = 5 - x ? (1) -x|x| > 0 (2) 5 - x > 0 Updated on: 18 Jan 2020, 06:38
Originally posted by gmatapprentice on 28 Dec 2019, 02:08.
Last edited by gmatapprentice on 18 Jan 2020, 06:38, edited 1 time in total.
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VeritasKarishma

Quick question on the following:

Is |x−5|=(5−x)?
or Is |5−x|=(5−x)?

How did you go from |x-5| to |5-x|? This is the only part that confuses me in the question.

I understand |x-5| can be expressed as x-5 or -(x-5)=5-x (without modulus)
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Is the square root of (x - 5)^2 = 5 - x ? (1) -x|x| > 0 (2) 5 - x > 0 29 Dec 2019, 23:21
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gmatapprentice
VeritasKarishma

Quick question on the following:

Is |x−5|=(5−x)|x−5|=(5−x) ?
or Is |5−x|=(5−x)|5−x|=(5−x)?

How did you go from |x-5| to |5-x|? This is the only part that confuses me in the question.

I understand |x-5| can be expressed as x-5 or -(x-5)=5-x (without modulus)


As long as you keep the absolute value sign, it doesn't matter if you flip what is inside. It stays the same only.


|x - 5| = |5 - x| in every case.

Try it. Put x = 1 or 0 or 6 or 10 or -4 etc. The absolute value will be the same in both cases. (5 - x) is just the negative of (x - 5) or (x - 5) is just the negative of (5 - x) so their absolute values will ALWAYS be the same.

They are both the same so when we remove the absolute value signs, we get the same result.

|x - 5| = (x - 5) when x >= 5
|x - 5| = -(x - 5) = (5 - x) when x < 5


|5 - x| = 5 - x when x <= 5
|5 - x| = x - 5 when x > 5

In both cases, the expression is equal to (x - 5) when x > 5 and the expression is equal to (5 - x) when x < 5 because that is how we will always get the positive value.
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Is the square root of (x - 5)^2 = 5 - x ? (1) -x|x| > 0 (2) 5 - x > 0 30 Dec 2019, 08:19
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VeritasKarishma many thanks this is clear now.
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Is the square root of (x - 5)^2 = 5 - x ? (1) -x|x| > 0 (2) 5 - x > 0 19 Feb 2020, 12:17
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Bunuel
dhushan
Just want to make sure that my line of thinking is right. I got D as well.

sqrt(x-5)^2 = 5-x
==> |x-5| = 5-x

which means that if

x-5<0, -(x-5) = 5-x = no solution, therefore x<5

x-5>0, x-5 = 5-x, which means that x>5 and x = 5

from (1) -x|x| > 0, we know that x has to be negative -(-x)|-x| is the only way to get a number greater than 0. Therefore, this means that of the three possible solutions for x, only this x<5 hold true.

from (2) 5-x>0, therefore x<5.

Can someone please point out if there is something wrong with my reasoning.

Is \(\sqrt{(x-5)^2}=5-x\)?

First of all, recall that \(\sqrt{x^2}=|x|\).

Is \(\sqrt{(x-5)^2}=5-x\)? --> is \(|x-5|=5-x\)? --> is \(x-5\leq{0}\)? --> is \(x\leq{5}\)?

(1) \(-x|x| > 0\) --> \(|x|\) is never negative (positive or zero), so for \(-x|x|\) to be positive, \(-x\) must be positive \(-x>0\) --> \(x<0\). Sufficient.

(2) \(5-x>0\) --> \(x<5\). Sufficient.

Answer: D.

Bunuel how did you get this part "is x−5≤0x−5≤0? --> is x≤5x≤5" from the previous eqn?
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