First of all, a review. If A and B are algebraic expressions, then the first step of solving
|A| = B
is to say
A = B or A = -B.
(BTW, the word "or" is an important piece of mathematical equipment; it's not garnish!)
Also, with absolute value equations, you always have to check your answers, because the algebraic approach can result in extraneous roots.
Now, we're ready to begin.
Prompt: Is x > 0?
Statement #1: |x + 3| = 4x - 3
We get:
x + 3 = 4x - 3 or x + 3 = -(4x - 3)
x + 3 = 4x - 3 or x + 3 = -4x + 3
6 = 3x or 5x = 0
x = 2 or x = 0
Check both:
|2 + 3| = 5
4(2) - 3 = 5 -----> x = 2 is a solution
|0 + 3| = 3
4(0) - 3 = -3 -----> x = 0 doesn't work: it's an extraneous solution.
Thus, statement #1 leads to the unambiguous answer of x = 2, so we know that x > 0. Statement #2, by itself, is sufficient.
Statement #2: |x - 3| = |2x - 3|
We get:
x - 3 = 2x - 3 or x - 3 = -(2x - 3)
x - 3 = 2x - 3 or x - 3 = -2x + 3
x = 0 or 3x = 6
x = 0 or x = 2
Check both answers.
|0 - 3| = 3
{2(0) - 3| = 3 -----> x = 0 works
|2 - 3| = 1
|2(2) - 3| = 1 -----> x = 2 works
This statement allows for two possible answers for x --- for one of them, x is > 0, and for the other it isn't. Therefore, this statement does not allows us to formulate a definitive answer to the prompt question. By itself, this statement is insufficient.
Answer = A
Does all this make sense?
Here's a related DS for further practice.
https://gmat.magoosh.com/questions/966
When you submit your answer to that question, the next page will have the video explanation.
Let me know if you have any questions.
Mike
