Please explain my answer in Proof Inequalities

[phillyolly]

Homework Statement

I solved the problem and it matches the answer at the end of the book.
Please explain why am I right? Why these two answers are correct?
THank you

Homework Equations

The Attempt at a Solution

 

[statdad]

You simply got extremely lucky with your first work - it is never correct to simplify

$$|x+1| + |x-2| = |x|+1 + |x| -2$$

as you did.

 

[phillyolly]

Do you imply that my approach is incorrect and need to be changed? Is it only my first part incorrect? The second one is alright?

 

[Mentallic]

You need to think of a few cases here.

1) When x+1 AND x-2 > 0, so that leaves x>-1. When you make this assumption you can get rid of the absolute value signs since it is, by your assumption, more than zero anyway so nothing changes.

2) When x+1>0 and x-2<0, so we restrict x to -1<x<2. Since one of the absolute values are less than zero, you need to take the negative of it when removing the absolute value sign.

3) When both are less than zero.

And you can include when x=-1,2 where necessary. Remember that since in each case, you make an assumption on what x is. When you solve the problem for that case, if x turns out to be something outside of the domain of your assumption, then that answer is invalid. For example, in case two if you end up solving the problem and get x=4, then the answer isn't valid since we already assumed for that case, -1<x<2.