Is x > 0?

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

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Bunnel,
For statement one, I understand that you have not solved for x in your solution. I am still learning the abs value and in equalities. Can you share your thoughts on the following solution (solving for x in both statements)

a. |x+3| = 4x -3
|x+3| = x + 3 for x + 3 >= 0 ---> x > -3.
Solving for x,
x + 3 = 4x -3 ---> 6 = 3x ---> x = 2 ( solution accepted since x > -3)

| x + 3| = -(x +3 ) for x + 3 < 0 --> x < -3.
Solving for x,
-x - 3 = 4x -3
5x = 0 ---> x = 0 ( solution discarded as x is not < -3)

A is sufficient.

b. |x -3| = |2x -3|
|x -3| = x -3 for x-3 >= 0 ---> x >= 3
|x -3| = -(x -3) for x-3 < 0 ---> x < 3

|2x -3 | = 2x -3 for 2x -3 >= 0 ----> x >=3/2
|2x -3 | = -(2x -3) for 2x -3 < 0 ----> x < 3/2

So we have 3 ranges

1) x >=3 ==> x - 3 = 2x -3 ====> x = 0 ; discard
2) 3/2 < x < 3 ===> -(x-3) = 2x -3 ====> -x + 3 = 2x -3 ===> x =2 ( accepted solution )
3) x < 3/2 ===> -(x-3) = -(2x -3) ====> x = 3 (discarded the solution)


Let me know if I have solved the question correctly. I know the process is lengthy.

One more question, if we consider the only acceptable solution in statement b i.e., x =2 (case 3/2 < x < 3), then both statements are sufficient independently and we get the answer as option (D).

Please discuss.

Regards

Works. Thanks for pointing out the mistake and quick response too. I am just learning this so working ground up, hence I am not hesitant in taking long route initially. After next few questions, I would be tuned to extract the ranges I guess. Is there a condition on squaring the sides of the equation or inequation.

Regards

Well, I have a little bit another solution for this problem

Consider (1): if x>-3 then x+3=4x-3, so x=2, sufficient
if x<-3, then -x-3=4x-3, which Leeds to answer x=0, which is wrong because we consider only x<-3
Sum up (1) is sufficient

The saim logic can be applied to the (2), and it is possible to derive that (2) is sufficient

So the answer is D either (1) is sufficient or (2) is sufficient

In your logic, you gave a mistake, because you've missed the root.

Posted from GMAT ToolKit

A simple solution for this is solve for x in each statement, but remember when you solve for x with abs value you must make the term positive and negative. in both cases you will find x = +# and zero. plug the numbers back into the equation and you will find zero will not fit the equation so x must be > 0
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Is this a correct way to approach 2) |x – 3| = |2x – 3| ?
Step 1:
(x-3) = (2x-3)
=> x = 0
Substitute x = 0 in |x – 3| = |2x – 3| => VALID

Step 2:
(x-3) = -(2x-3)
=> x = 2
Substitute x = 2 in |x – 3| = |2x – 3| => VALID

So 2) |x – 3| = |2x – 3| is NOT SUFFICIENT

Pretty slick answer. Saves a lot of time.

If you substitute x=0 in |x + 3| = 4x – 3 you'll get: LHS=|x + 3|=3 and RHS=4x – 3=-3, thus LHS\neq{RHS}, which means that x=0 is not the root of the given equation.

When expanding |x+3|:

When x<-3, then |x+3|=-(x+3), so in this case we'll have -(x+3)=4x-3 --> x=0 --> discard this value since 0 is not less than -3 (we consider the range when x<-3).

When x>=-3, then |x+3|=x+3, so in this case we'll have x+3=4x-3 --> x=2 --> this value of x is OK since 2>-3.

So, |x + 3| = 4x - 3 has only one root, x=2.

Hope it's clear.
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