lnm87 wrote:
If |x - 5| = 2|x - 8|, then what is the value of x?
(1) \(|x^2 – 100| > 50\)
(2) \(|x^2 – 49| =0\)
I've got some math in the question stem, so I'll start by writing it down and then simplifying it.
|x - 5| = 2|x - 8|
I like to think of absolute values containing subtraction as being "distances". For example, |x - 5| is the distance between x and 5 on a number line, and |x - 8| is the distance between x and 8 on a number line.
So, this question stem says "the distance between x and 5 on a number line is twice the distance between x and 8 on a number line."
Or in other words, "x is twice as far from 5 as it is from 8."
Sketch a number line to see where this will happen:
-----------5---------x----8-------
-----------5--------------8--------------x--------
A little guessing and testing confirms that x = 7 and x = 11 are the two cases where this is true.
So, the question stem says "either x = 7, or x = 11." That's a much easier translation. It also makes the statements easier to work with: we only ever need to test the values 7 and 11! Now on to the statements.
Statement 1: Given that |x^2 - 100| is greater than 50, can I exclude either 7 or 11?
Yes. x can't equal 11, because |11^2 - 100| is less than 50. And it can't be any other number, aside from 7, since the question stem says that x is definitely either 7 or 11. Therefore, x = 7, and the statement is
sufficient.
Statement 2: This lets me exclude x = 11 as well. Therefore, x = 7, and the statement is
sufficient.
The correct answer is
D.