Bunuel wrote:
What is the value of x?
(1) |x + 9| = 2x
(2) |2x − 9| = x
Statement 1:Since the right side is 2x = An absolute value, we can guarantee \(2x >= 0\). Then \(x + 9\) must be positive. This simplifies to \(x + 9 = 2x\) and there is only one solution. Sufficient.
Statement 2:Again we must have \(x >= 0\). Yet this doesn't tell us if \(2x - 9\) is positive or negative so we need to split into two cases, \(0 <= x < 4.5\) or \(x >= 4.5\).
First one gives \(9 - 2x = x\) and \(x = 3\) which is within range of {0. 4.5), so it is a viable solution.
The second one gives \(2x - 9 = x\) and \(x = 9 >= 4.5\), which is within range so it is another viable solution.
Hence we have two different solutions, insufficient.
Ans: A
Another way to "simplify" statement 2 is to square it to get \(4x^2 - 36x + 81 = x^2\) and \(x^2 - 12x + 27 = 0\). We get x = 3 or x = 9 but we have to plug these back in to check if they work.
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