Is |x| < 1 ?

[NMencia09]

Is |x| < 1 ?

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

(2) |x - 3| > 0

Prompt asks: <---(-1)---(0)---(1)--->
<----o=========o---->?

For (2): Is x more than 3 units away from 0 on a number line?
<---(-3)----(0)-----(3)---->
<===============o====>
Insufficient.

For (1) INSUFFICIENT: (PER EXPLANATION by MGMAT) There are three possible equations here if we open up the absolute value signs:

1. If x < -1, the values inside the absolute value symbols on both sides of the equation are negative, so we must multiply each through by -1 (to find its opposite, or positive, value):

|x + 1| = 2|x -1| -(x + 1) = 2(1 - x) x = 3
(However, this is invalid since in this scenario, x < -1.)

2. If -1 < x < 1, the value inside the absolute value symbols on the left side of the equation is positive, but the value on the right side of the equation is negative. Thus, only the value on the right side of the equation must be multiplied by -1:

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

3. If x > 1, the values inside the absolute value symbols on both sides of the equation are positive. Thus, we can simply remove the absolute value symbols:

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

Thus x = 1/3 or 3. While 1/3 is between -1 and 1, 3 is not. Thus, we cannot answer the question.


I'm confused as to how this process works. Can anyone help here?
1) Why do 3 scenarios pop up for statement (1)? I guess you are looking at the values that make 1 Absolute value negative, and then seeing how that range of values affects all other Absolute values in the problem?
a. find the values which make an abs. value negative.
b. multiply by -1 any other abs values which are made negative by the first range of values.
c. solve for variable (x)
d. check to see if variables lies within the range.
e. if so, then what? How do you relate this back to the original problem?

How do you know to include -1< x < 1?

thanks...

[arnabgangully]

please solve the |x| by substituting the values of -x and +x and then open up you will get three solutions

[NMencia09]

Maybe I'm making this too complicated....?

[jnelson0612]

Hi Noah,
We've now gone over this together in office hours, so I think you're good, but to answer your question for the benefit of anyone reading this question, we look at three scenarios because if we rephrase the original question we find that it is really asking whether x is between 1 and -1. So we have three possible scenarios:
x is less than -1 (answer to the question is NO)
x is between -1 and 1 (answer to the question is YES)
x is greater than 1 (answer to the question is NO)

So discovering where x falls on the number line is critical to answering this question.

[sachin.w]

I have a doubt here..

|x - 3| > 0

if we simplify this,

x-3 > 0 OR -(x-3) >0

x>3 OR x<3


This does say that x is not between -1 and 1.

So B must be sufficient.
Please correct me if I am wrong here.

[RonPurewal]

x>3 OR x<3


This does say that x is not between -1 and 1.

nope.
look again at what you wrote: "x > 3 OR x < 3" (which is perfectly correct, by the way).
this statement allows x to be any value in the world, except 3.
some of those values are between -1 and 1; others aren't. not sufficient.

[sachin.w]

Thanks a lot Ron. Its very much clear now.

[tim]

:)