What is the value of |x|?

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

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

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

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.

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

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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