No properties - Modulus

[rahcool4u]

Can anyone explain how to solve the below modulus questions (from gmat prep).

Q1) Is √(x-3)^2 = 3 –x ?

1) x ≠ 3
2) –x│x│ > 0

Q2) Is │x│ = y – z ?

1) x +y =z
2) x < 0


Answers:
Q1) B
Q2) C

[Robinmrtha]

Q1) Is &#8730;(x-3)^2 = 3 –x ?

&#8730;(x-3)^2 = 3 –x
For this condition to be true...x should always be less than 3
So, the question is asking if x is greater than or less than 3

statement 1
x &#8800; 3
So insufficient as x can be both greater than or less than 3

statement 2
–x&#9474;x&#9474; > 0
for the left hand side to be true x should be negative...
so x is negative hence it is less than 3
sufficient...
Answer B

[Robinmrtha]

Q2) Is &#9474;x&#9474; = y – z ?

1) x +y =z
2) x < 0

The question is asking whether y is greater than z???
statement 1
x+y=z
or y =z-x
y is greater than z if x is positive and less than z if x is negative...
so insufficient

statement 2
x is negative but does not say anything about y or z
insufficient

when combined since x is negative y is less than z
sufficient
hence answer is C

[rahcool4u]

Thanks Robinmrtha
Q1 is still not clear to me.

Why the condition &#8730;(x-3)^2 = 3 –x , is true only for -ve values of x.
eg:

Take x =5, &#8730;(5-3)^2 = 3 –5 -> 2 = -2 -> condition 1 fails for +ve values
Take X =-5, &#8730;(-5-3)^2 = 3 +5 -> -8 = 8 -> condition 1 fails for -ve values
Take X =0, &#8730;(0-3)^2 = 3 -0 -> -3 = 3 -> condition 1 fails for 0

According to me, the answer should be D.
Did I miss something ?

[Robinmrtha]

Fren when you take
Take X =-5, &#8730;(-5-3)^2 = 3 +5 -> &#8730;64 = 8
you are doing the same mistake when you are taking x=0

Hope this helps

[carpe_diem]

Thanks Robinmrtha
Q1 is still not clear to me.

Why the condition &#8730;(x-3)^2 = 3 –x , is true only for -ve values of x.
eg:

Take x =5, &#8730;(5-3)^2 = 3 –5 -> 2 = -2 -> condition 1 fails for +ve values
Take X =-5, &#8730;(-5-3)^2 = 3 +5 -> -8 = 8 -> condition 1 fails for -ve values
Take X =0, &#8730;(0-3)^2 = 3 -0 -> -3 = 3 -> condition 1 fails for 0

According to me, the answer should be D.
Did I miss something ?

When you take X = -5 and do &#8730;(-5-3)^2 then this would be &#8730;64 = 8 and not -8. This is because in GMAT &#8730;X will always be positive. So you need to first square the number 8 which is 64 and then take square root which in this case will always be + ve. Hence 8 = 8. Sufficient.

[mehravikas]

Please correct me if I am wrong - Can I simplify the question to be:

&#8730;(x-3)^2 = 3 -x

x - 3 = 3 - x
i.e. 2x = 6
Is x = 3???

Fren when you take
Take X =-5, &#8730;(-5-3)^2 = 3 +5 -> &#8730;64 = 8
you are doing the same mistake when you are taking x=0

Hope this helps

[aj5105]

IMO, No.

Sqrt(x^2) = |x| . So, simplify the question on those terms.

Please correct me if I am wrong - Can I simplify the question to be:

&#8730;(x-3)^2 = 3 -x

x - 3 = 3 - x
i.e. 2x = 6
Is x = 3???

Fren when you take
Take X =-5, &#8730;(-5-3)^2 = 3 +5 -> &#8730;64 = 8
you are doing the same mistake when you are taking x=0

Hope this helps

[aj5105]

Is &#8730;(x-3)^2 = 3 –x ?

1) x &#8800; 3
2) –x&#9474;x&#9474; > 0


IMO, the stem statement is asking if x = 3 or x is greater than 3.

Statement 1 : We cannot say. Insufficient.

Statement 2 : We know x < 0. Definite No. So (B)

Please correct me if I am wrong.

[mehravikas]

Can you elaborate further by giving another example please?

IMO, No.

Sqrt(x^2) = |x| . So, simplify the question on those terms.

Please correct me if I am wrong - Can I simplify the question to be:

&#8730;(x-3)^2 = 3 -x

x - 3 = 3 - x
i.e. 2x = 6
Is x = 3???

Fren when you take
Take X =-5, &#8730;(-5-3)^2 = 3 +5 -> &#8730;64 = 8
you are doing the same mistake when you are taking x=0

Hope this helps

[cramya]

IMO, the stem statement is asking if x = 3 or x is greater than 3.

Statement 1 : We cannot say. Insufficient.

Statement 2 : We know x < 0. Definite No. So (B)

Please correct me if I am wrong.

Replying to PM and building upon the explanation:


sqrt(X^2) = |x| as u pointed out rightly Similarly in this prob sqrt[(x-3)^2] = |x-3| here our X = x-3

The question is asking Is |x-3| = 3-x True when x<=0 False when x>0

Stmt I

x &#8800; 3

Cant say for sure if x <=0 or x>0

INSUFF

Stmt II

-x |x| > 0

|x| cannot be 0 since -x |x| >0

By defnition |x| is x when x>=0 and |x| = -x when x<0

You can either apply the defnition or pick a negative and positive number and see what happens

Picking numbers is easy here rather than getting tangled with the theory(IMO):

Let x = 2

- (2) |2| > 0 NO

Let x=-2

- (-2) * |-2|

2 * 2 >0 TRUE


-x |x| > 0 only when x<0 in this prob so x is neagtive therefore
&#8730;(x-3)^2 = 3 –x

B

Hope this helps!

Regards,
CR