Is x > 0?

[PGMAT]

Is x > 0?

(1) |x + 3| = 4x - 3
(2) |x - 3| = |2x - 3|

[Mike@Magoosh]

Hi, there. I'm happy to help with this.

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

[sanju09]

Superb Mike@Magoosh!!

[ronnie1985]

Thank you... Need to do more practice in mod problems

[CharuB]

Hi,

I have a little confusion regarding Absolute values.

In the question Statement#2 has absolute values on both the sides,So would it be correct if i say we get:
x - 3 = 2x - 3 or -(x - 3) = -(2x - 3)
Please correct me if i am wrong.

kindly let me know if the correct answer to the question would be choice(E).

Thank you.

[Whitney Garner]

Hi,

I have a little confusion regarding Absolute values.

In the question Statement#2 has absolute values on both the sides,So would it be correct if i say we get:
x - 3 = 2x - 3 or -(x - 3) = -(2x - 3)
Please correct me if i am wrong.

kindly let me know if the correct answer to the question would be choice(E).

Thank you.

Hi CharuB!

I can definitely sympathize with the temptations to do that, but there are actually 4 cases we would have to consider (because the expression on the right could be +/- and the expression on the left could be +/- and they don't have to be - or + at the same time. So technically, it could be any of the following:

x - 3 = 2x - 3
-(x - 3) = -(2x - 3)
-(x - 3) = 2x - 3
x - 3 = -(2x - 3)

BUT - notice that some of these are actually identical:

x - 3 = 2x - 3
-(x - 3) = -(2x - 3)
All we have done is multiply both sides by a -1, so they are the same equations!

-(x - 3) = 2x - 3
x - 3 = -(2x - 3)
And again, all we have done is multiply both sides by a -1, so they are the same equations!

So technically, to solve we just need to examine 2 cases:
x - 3 = 2x - 3
x - 3 = -(2x - 3)

Hope that clears up the question!

Whit

[CharuB]

Thanks Whitney.

[vikram4689]

trick here is whenever we have single || then we have to put the value of variable back in equation and see whether our solution satisfies the equation but for double || cases, we don't have to check by putting value in equation. whatever we found by solving the equations will eventually be solution.

both i) and ii) give x=0,2 but x=0 does not satisfy i) so x=2>0 is proved. for ii) x=0,2 hence insuff.