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

[Guest]

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

1) -x|x|>0
2)5-x>0

Ans- D ( BOTH SUFFICIENT)

Source: Gmatprep 2

Please explain?

[Guest]

Hey RON-

I have an exan next week. ur explanation here would be great help!

[mbarshaik]

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

Hope this helps you. All teh best for your exam.

[RonPurewal]

this is an excellent explanation, to which i'll add only a few comments.

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

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

general takeaway here:
squaring a quantity, and then square-rooting, is equivalent to taking the absolute value.
remember this.

|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

we can make this more clear:
|x - 5| can be either (x - 5), the actual quantity within the absolute-value bars, or (5 - x), the opposite of that quantity.
if it's to be the original quantity (x - 5), then that quantity must be at least 0: x > 5.
if it's to be the opposite (5 - x), then that opposite quantity must be at least 0. for that to happen, x < 5.
(notice that, if x is actually 5, then |x - 5| equals both (x - 5) and (5 - x), since both of them are zero.)

therefore, we can rephrase the question:
is x < 5?

1. -x|x| > 0

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

when you see this statement, it may bewilder you at first, but you should look at it and think: "ok, just absolute-value bars and negative signs. no other numbers; no other operations; this could only possibly have to do with the sign of x."
then just test it to see whether it works for PNZ (positive, negative, zero).
turns out that it only works for negative numbers.
therefore, rephrase:
(1) x < 0

this is sufficient, since x is definitely less than 5 if it's negative.

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

nothing to add.

[veronica.tong]

Ron,

Can you please explain what you mean by:

if it's to be the original quantity (x - 5), then that quantity must be at least 0: x > 5.
if it's to be the opposite (5 - x), then that opposite quantity must be at least 0. for that to happen, x < 5.
(notice that, if x is actually 5, then |x - 5| equals both (x - 5) and (5 - x), since both of them are zero.)

Why must the left side of the equation be at least zero? Where did you get that information from?

Been reading your explanation over and over again, I get everything else except for that part. I understand that if you plug in numbers, like -6 vs. 6, the answer is very obvious, but I want to know how you can solve this problem without doing that. Please help! Thank you!

[RonPurewal]

Ron,

Can you please explain what you mean by:

if it's to be the original quantity (x - 5), then that quantity must be at least 0: x > 5.
if it's to be the opposite (5 - x), then that opposite quantity must be at least 0. for that to happen, x < 5.
(notice that, if x is actually 5, then |x - 5| equals both (x - 5) and (5 - x), since both of them are zero.)

Why must the left side of the equation be at least zero? Where did you get that information from?

Been reading your explanation over and over again, I get everything else except for that part. I understand that if you plug in numbers, like -6 vs. 6, the answer is very obvious, but I want to know how you can solve this problem without doing that. Please help! Thank you!

absolute values must be at least 0.

consider a simpler expression: just |x|.
this expression is equal to x, whenever x is at least 0.
however, this expression is equal to -x, whenever -x is at least 0 (i.e., whenever x < 0).

the same sort of thing is happening here, except now it's (x - 5) instead of just x, and (5 - x) instead of just -x.

does that help?

[sandeep.19+man]

Hi Ron,

Under normal circumstances, I would arrive at
|x-5| = 5-x and then try to solve for x

Case 1: |x-5| is +ve

x-5 = 5-x

2x = 10

x = 5

Case 2: |x-5| is -ve

-(x-5) = 5-x

5-x = 5-x

no solution

At this stage i will be stuck. As I will try to rephrase the question as
Is x = 5 or is x = ? (no solution).

This is the way we normally solve for equations to find possible values for x. What is wrong with this approach?

[tim]

Bad idea in general. You should never "arrive at" |x-5|=5-x and then solve for x using the statements. |x-5|=5-x is the question that is being asked. You can simplify the equation in the question, but don't ever take it as a given and try to analyze the statements. This is exactly the opposite of what you should do. Always take the information in the statements in the given and try to answer the question, NOT vice versa..

[supratim7]

general takeaway here:
squaring a quantity, and then square-rooting, is equivalent to taking the absolute value.
remember this.

Thank you so much for this takeaway.. Very helpful.

|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

we can make this more clear:
|x - 5| can be either (x - 5), the actual quantity within the absolute-value bars, or (5 - x), the opposite of that quantity.
if it's to be the original quantity (x - 5), then that quantity must be at least 0: x > 5.
if it's to be the opposite (5 - x), then that opposite quantity must be at least 0. for that to happen, x < 5.
(notice that, if x is actually 5, then |x - 5| equals both (x - 5) and (5 - x), since both of them are zero.)

therefore, we can rephrase the question:
is x < 5?

But I am struggling to internalize this idea/working..
Are you simply saying something like this??

Question: Is |x-5| = 5-x ?

Logic: Because "5 - x" = "absolute value of something", "5 - x" at the least can be 0 (i.e. it cannot be less than 0)

So, rephrase: Is 5 - x ≥ 0? i.e. x ≤ 5?

Many thanks | Supratim

[jnelson0612]

That is absolutely right, supratim. Very good. :-)

[supratim7]

Great.. Thank you Jamie :)

So can we generalize following?

if the question asks "Is | some expression with variables | = other expression with SAME variables?", go ahead and rephrase it as "Is other expression with SAME variables ≥ 0?"

For example..
Is |x - 5| = 5 - x ?
Is 5 - x ≥ 0? The rephrase would work in this case

Is |a - 5| = 5 - b ?
Is 5 - b ≥ 0? The rephrase wont work in this case. Right?

I am just trying to arrive at a valid takeaway..

Many thanks | Supratim

[tim]

No, it doesn't have to do simply with having the same variables. Instead the generalized form would be:

is |expression| = -expression ?

then the rephrased question is whether expression is less than or equal to 0

[supratim7]

I am struggling a lot with this one.. guess my concepts are not sound.. several queries..

(A) I can grasp following..

we can make this more clear:
|x - 5| can be either (x - 5), the actual quantity within the absolute-value bars, or (5 - x), the opposite of that quantity.
if it's to be the original quantity (x - 5), then that quantity must be at least 0: x > 5.
if it's to be the opposite (5 - x), then that opposite quantity must be at least 0. for that to happen, x < 5.
(notice that, if x is actually 5, then |x - 5| equals both (x - 5) and (5 - x), since both of them are zero.)

But can't grasp the TRANSITION to

therefore, we can rephrase the question:
is x < 5?

(B) I took following approach. Is it OK?

Is √{(x-5)^2} = 5-x ?
1) -x|x| > 0
2) 5-x > 0

Is √{(x-5)^2} = 5-x ?
so, Is |x-5| = 5-x ?

1) -x|x| > 0
so, x < 0
any negative number satisfies |x-5| = 5-x

2) 5-x > 0
so, x < 5
any number less than 5 satisfies |x-5| = 5-x

Ans: D

(C) If the question is "Is √{(x-5)^2} = 5-x ?", Why cant we square both side and solve? i.e.

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

(D) What is the correct rephrase of "Is |x| = y?"

"Is x = y? and x ≥ 0? or Is -x = y? and x ≤ 0?"
OR
"Is x = y? and x > 0? OR Is -x = y? and x < 0?"

Thank you in advance

[RonPurewal]

1) -x|x| > 0
so, x < 0
any negative number satisfies |x-5| = 5-x

2) 5-x > 0
so, x < 5
any number less than 5 satisfies |x-5| = 5-x

If you understand the red part here, then you probably also understand my rephrasing of the question. It's the same concept.

I.e., you seem to understand that |x - 5| = 5 - x is true if x < 5, but false if x > 5.
If you understand that, that's the key to the rephrasing I made above. Let it sink in for a second.

[RonPurewal]

(C) If the question is "Is √{(x-5)^2} = 5-x ?", Why cant we square both side and solve? i.e.

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

This doesn't work for the same reason "squaring both sides" doesn't work in general.
Consider a simpler example:
-3 = 3
That's false, right?
If you could "square both sides" then you would erroneously claim that -3 = 3 is true, because 9 = 9 is true.

Same issue here. If you "square both sides", you lose the ability to tell whether the two original expressions were actually the same, or opposites.