What is the value of |x|?

Question

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

(2) x^2 = 8x - 16

shasadou · Accepted Answer

OA:

Note that the question is asking for the absolute value of x rather than just the value of x. Keep this in mind when you analyze each statement.

(1) SUFFICIENT: Since the value of x2 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: 
|x2 + 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.

(2) SUFFICIENT: 
x2 = 8x – 16 
x2 – 8x + 16 = 0 
(x – 4)2 = 0
(x – 4)(x – 4) = 0 
x = 4

Therefore, |x| = 4; this statement is sufficient.

The correct answer is D.

BudhrajaAcademy · Answer

The answer should be option (2) - statement 2 alone is sufficient but statement (1) alone is not sufficient. Here's the solution:

(1) Using Statement 1 alone,

|x^2 + 16| - 5 = 27

It means that either (x^2 + 16) - 5 = 27 or (x^2 + 16) - 5 = - 27

Solving the first of these, x^2 + 16 = 32 => x^2 = 16 => x = -4 or 4

Solving the second, x^2 + 16 = -22 => x^2 = -6 => x = \sqrt{6}, -\sqrt{6}

Using statement 1 alone, we don't get a unique answer.

Using Statement 2 alone,

x^2 = 8x - 16 => x^2 - 8x + 16 = 0 => x^2 - 4x - 4x + 16 = 0 => (x-4) (x-4) = 0

This gives x = 4 and |x| = 4.

So, statement 2 alone is sufficient but statement 2 alone is not sufficient

DangerPenguin · Answer

This was my rationale for picking choice D

statement 1: sufficient
After adding 5 to each side you can set x^2 + 16= -32 and x^2 + 16 = 32 since it has absolute value brackets around it. Thus, x = 4 or x = -4, and the absolute value of each of those values is 4.

statement 2: sufficient
 - I subtracted 8x-16 from x^2 to get x^2 - 8x + 16 = 0
 - Factored to get (x-4)(x-4)
 - x =4 which has absolute value of 4

bankerboy30 · Answer

D is correct, the absolute value portion of the equation will never be negative as X^2 is always positive

kinghyts · Answer

Statement 1:

|x^2 + 16| - 5 = 27 
|X^2+16| = 32

Since X^2 will always be positive
x^2 = 16
therefore |x| = 4. Sufficient

Statement 2:

x^2 = 8x - 16
(x-4)^2=0
x = 4 , therefore |x|=4. Sufficient

D) should be the answer

MathRevolution · Answer

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

What is the value of |x|?

(1) |x^2 + 16| – 5 = 27 
(2) x^2 = 8x – 16
 
There is 1 variable (x) in the original condition. In order to match the number of variables and the number of equations, we need 1 equation. Since the condition 1) and 2) both has 1 equation, there is high chance D is going to be the answer.

In the case of the condition 1), we can obtain, |x^2+16|=5+27=32. Then, |x^2+16|=-32 or 32. However, since x^2+16=-32 is only possible when x^2=-48, this is not possible.
In case of x^2+16=32, x=-4 or 4. However, since |x|=|-4|=|4|=4, the answer is unique and the condition becomes sufficient.

In the case of the condition 2), we can obtain x^2-8x+16=0. Then, (x-4)^2=0. So, x=4. Since, |x| can be 4, the answer is unique, and the condition is sufficient. Therefore, the correct answer is D.

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously, there may be cases where the answer is A, B, C or E.

mvictor · Answer

since we are asked for the absolute value of x, we need to find the value of either x or -x.

1. |x^2+16|=32
ok 2 options:
x^2 +16=32 => x^2 = 16 -> |x|=4
x^2 +16 = -32 -> x^2 = -48 - this is not possible
only one option: |x|=4. sufficient.

2. x^2 -8x+16 = 0
(x-4)(x-4)=0
x=4
|x|=4. sufficient.

D

dcummins · Answer

I don't know that this has been explained x^2 = -48 is not a possible solution for Statement 1, so x= 4 and x=-4 are correct solutions, both lead to the same solution when in absolute value terms

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What is the value of |x|?

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