What is the difference between the maximum and the minimum values of x?

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[GMAT math practice question]

What is the difference between the maximum and the minimum values of x?

1) x satisfies 2 < \(\sqrt{|x-2|}\) < 4.
2) x is an integer.

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Since we have 1 variable (x) and 0 equations, D is most likely the answer. So, we should consider each condition on its own first.

Condition 1)

Since we have 2 < \(\sqrt{|x-2|}\) < 4, we have 4 < | x – 2 | < 16, by squaring everything.

Two Cases
First Case – positive value
(x – 2) = 4
x = 6

(x – 2) = 16
x = 18
Then we have 6 < x < 18

Second Case – negative value
-(x – 2) = 4
-x + 2 = 4
-x = 2
x = -2

-(x - 2) = 16
-x + 2 = 16
-x = 14
x = -18
Then we have -14 < x < -2

It means we have -14 < x < -2 or 6 < x < 18.
However, we don’t have either a maximum value of x or a minimum value of x.

Since we can’t specify a unique solution, it is not sufficient.

Condition 2)

Since condition 2) does not yield a unique solution, it is not sufficient.


Conditions 1) & 2)
Using our solutions from earlier, namely -14 < x < -2 or 6 < x < 18, we know that the possible values of x satisfying both conditions are -13, -12, … , -3 and 7, 8, … , 17.
The maximum value of x is 17 and the minimum value of x is -13.
The difference between the maximum and the minimum value of x is 17 – (-13) = 30.

Therefore, C is the answer.
Answer: C

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A, B, C, or E.