Is sqrt((x-3)^2) = 3 - x?

[niitorious]

Is sqrt((x-3)^2) = 3 - x?

1) x is not equal to 3
2) -x|x| > 0

I don't think I even understand what the question is asking. Is it possible to just cancel out the sqrt sign with the square? I thought you shouldn't do that because it leads to loss of information?

[abhinav_iitg]

Is sqrt((x-3)^2) = 3 - x?

1) x is not equal to 3
2) -x|x| > 0

I don't think I even understand what the question is asking. Is it possible to just cancel out the sqrt sign with the square? I thought you shouldn't do that because it leads to loss of information?

If you remember the following rule then the rest is piece of cake:

sqrt(x^2) = |x| as square root of a number is always positive.

for above question it is |x-3|. now the question is when |x-3| = 3 - x
clearly when x <= 3.

[RonPurewal]

good post above.

i'll elaborate on the following:

now the question is when |x-3| = 3 - x
clearly when x <= 3.

i don't know if i'd go as far as to say "clearly"; this is actually very difficult for most students of the gmat.

takeaway:
the absolute value will do one of two things to a quantity:
(a) LEAVE THE QUANTITY ALONE, if the quantity is POSITIVE;
(b) REVERSE THE SIGN of the quantity, if the quantity is NEGATIVE.
if the quantity is exactly 0, then both of these result in the same number, so it doesn't matter which of them you call it.

therefore:
the expression |x - 3| will equal one of two expressions:
LEFT ALONE as (x - 3), if x - 3 is POSITIVE -- i.e., if x is greater than 3;
REVERSED to (3 - x) (which is the same as -x + 3), if x - 3 is NEGATIVE -- i.e., if x is less than 3;
EITHER of these (since both equal 0) if x is exactly 3.

therefore, we now have a rephrase of the question.
REPHRASE:
is x ≤ 3 ?

so, the answer is (b).

[RonPurewal]

you can also solve this problem, perhaps more easily, by PLUGGING IN NUMBERS.

statement (1):
since 3 is clearly a pivotal number in this problem, try numbers that are greater than 3 and numbers that are less than 3.

try x = 0:
√((x - 3)^2) = 3
3 - x = 3
answer to prompt question = YES

try x = 5:
√((x - 3)^2) = 2
3 - x = -2
answer to prompt question = NO

insufficient.

--

statement (2)
this statement is an obnoxious way of stating that x is a negative number. (you still have to figure that out -- no way around it)

if you try plugging in a vast array of negative numbers - big, small, even, odd, etc. - you'll find that the equality in the prompt question holds for ALL of them.

sufficient.

[ashish.jere]

for (2) to be true, x has to be negative.

[RonPurewal]

for (2) to be true, x has to be negative.

true.

this is what i said in the post directly above yours:

statement (2)
this statement is an obnoxious way of stating that x is a negative number.

did you see that post?

if you did, are you trying to say something here in response, or was there something you didn't understand?

[tomslawsky]

Ron,
I understand the mechanics of your answer until you made the leap to is X<=3. Can you please briefly elaborate? Thanx.

[RonPurewal]

see my comments as "editor:" in the previous post

-- ron

[tomslawsky]

I COMPLETELY missed the word "sqrt", now the question seems very simple. Man, my lack of READING the question can really kill me sometimes. Thank you guys!

[Ben Ku]

Glad it was helpful!

[ishkaran88]

Thanks to Manhattan Staff and Abhinav.
You cleared a major flaw in my approach!

Regards
Ishkaran

[zchampz]

god post above.

i'll elaborate on the following:

now the question is when |x-3| = 3 - x
clearly when x <= 3.

i don't know if i'd go as far as to say "clearly"; this is actually very difficult for most students of the gmat.

takeaway:
the absolute value will do one of two things to a quantity:
(a) LEAVE THE QUANTITY ALONE, if the quantity is POSITIVE;
(b) REVERSE THE SIGN of the quantity, if the quantity is NEGATIVE.
if the quantity is exactly 0, then both of these result in the same number, so it doesn't matter which of them you call it.

therefore:
the expression |x - 3| will equal one of two expressions:
LEFT ALONE as (x - 3), if x - 3 is POSITIVE -- i.e., if x is greater than 3;
REVERSED to (3 - x) (which is the same as -x + 3), if x - 3 is NEGATIVE -- i.e., if x is less than 3;
EITHER of these (since both equal 0) if x is exactly 3.

therefore, we now have a rephrase of the question.
REPHRASE:
is x ≤ 3 ?

I didn't get it ...how can we rephrase like this ....can you please elaborate?

[imanemekouar]

Ron , what do you mean by this sentence.
if the quantity is exactly 0, then both of these result in the same number, so it doesn't matter which of them you call it.

[joshua.higgins]

Am I missing something?

If you plug in these two numbers:
x=-1/2
x=-5

Statement 2:
-(-1/2)|-1/2|>0
1/4>0 True

-(-5)|-5|>0
25>0 True

Main question:
Is sqrt((x-3)^2) = 3 - x?

sqrt(((-1/2)-3)^2) = 3 - (-1/2)
3.5 = 2.5 (False)

sqrt(((-5)-3)^2) = 3 - (-5)
8=8 (True)

I generally tend to make pretty inane errors with things like this, which is totally possible here, but I've now done this like 5 times and I keep coming back to the same issue....

For this to work, wouldn't x need to be an integer? (which isn't specified)

What's the deal?

[RonPurewal]

@ joshua.higgins

sqrt(((-1/2)-3)^2) = 3 - (-1/2)
3.5 = 2.5 (False)

your mistake is on the right side here. remember that subtracting a negative is equivalent to adding a positive.
therefore, 3 - (-1/2) is 3 + 1/2, or 3.5. so this is also "true".