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.

#1
|x + 9| = 2x
solve we get x=9 & -3
LHS cannot be -ve so -3 is not possible x=9 sufficient
#2
|2x − 9| = x
x=9,3
two values of x insufficient
option A is correct

Hi Bunuel,
I have a question about the solution you mentioned above. Why did we straight away take positive value for the first statement and not for the second one? The second statement has the same structure as the first one.

|2x -9 | = x, what if I move forward the same way as in statement 1. That is: x is positive and |2x-9| is an absolute value, hence always positive. Therefore
> |2x - 9| = x
> 2x - 9 = x
> x = 9.

What am I getting wrong here?

You are right to deduce that |2x -9 | = x implies that x must be more than or equal to 0. But |2x - 9| = 2x - 9 ONLY IF 2x - 9 >= 0 and x >= 0 does not necessarily means that 2x - 9 >= 0. For example, if x = 1, then 2x - 9 = -7. So, for some positive values of x, 2x - 9 will be positive and for some positive values of x, 2x - 9 will be negative. This is why we cannot use the same technique for (2) as we used for (1).

Hope it's clear.