What is the value of |x|? (1) |x^2 + 16| â€“ 5 = 27

by **Adi** Sat Jul 28, 2007 1:17 pm

My answer to the following question was B. The answer in the explanation is D. I am not sure if I agree with the explanation for (1). Please see below. - My reasoning is in Bold.

What is the value of |x|?

(1) |x^2 + 16| - 5 = 27

(2) x^2 = 8x - 16

TEST RESPONSE:

1) SUFFICIENT: Since the value of x^2 must be non-negative, the value of (x2 + 16) is always positive, therefore |x2 + 16| can be written x2 +16. Using this information, we can solve for x:
|x^2 + 16| - 5 = 27
x2 + 16 - 5 = 27
x2 + 11 = 27
x2 = 16
x = 4 or x = -4

Since |-4| = |4| = 4, we know that |x| = 4; this statement is sufficient.

MY RESPONSE
1) INSUFFICIENT:
One of the solutions would be the one state above. Wouldn't this following one also be a solution for condition 1?
|x^2 + 16| - 5 = 27
x^2 - 16 - 5 = 27
X^2= 48 ....x=sqrt(48) ...condition 1 gives 2 answers hence insufficient.

What am I doing wrong?

by **a7lee** Sat Jul 28, 2007 1:26 pm

Abs( x^2 + 16 ) will always be positive. Since the x is square you don't have to worry about the equation inside the absolute value becoming negative.

Basically....x^2 + 16 > 0 always, no matter what you do.

If it was x + 16 then its possible that it could be less than 0. So you would have to account for the -x - 16 equations when x < -16.

by **cindyqtran** Sat Jul 28, 2007 2:43 pm

You had x^2 - 16 - 5 = 27 instead of x^2 + 16 - 5 = 27

|x| = 4

by **StaceyKoprince** Sat Jul 28, 2007 6:04 pm

Your error is in the following:

|x^2 + 16| - 5 = 27
x^2 - 16 - 5 = 27

As Cindy said, when you remove the absolute value signs, you do not change the sign in front of the 16 to negative. It stays the same.

And, as a7lee said, you would have to check the negative version if we didn't have a square here, but even then you would have made an error - you don't just make the 16 negative. The entire expression needs to be made negative. It would have been:
|x^2 + 16| - 5 = 27
|x^2 + 16| = 32
x^2 + 16 = 32 AND x^2 + 16 = -32
We can solve the first equation but the second equation can't be solved (well, on the GMAT anyway - we'd have to use imaginary numbers, which isn't allowed on the GMAT).