How to solve an inequality that apparently has more than one case?

[hatsie3]

Hey guys, My prof told me that this inequality has 6 cases. I have no clue how to start. It is below:

abs(x-1) +2abs(x) - abs(x+1) + abs(x-2) >= abs (x-4)


I attached the question below just in case the way I wrote it was not clear enough.

 

[mmm4444bot]

When we're given the absolute value of a symbolic expression, we often need to consider both cases: (1) the symbolic expression represents a non-negative number, or (2) it represents a negative number.

That is, in the expression |x-1|, the symbolic expression x-1 could represent x-1 or it could represent 1-x.

I see only five absolute value expressions, so I'm not sure why your instructor referred to six cases.

 

[mmm4444bot]

PS: You're not necessarily looking for five (or six) solutions. The solutions to this exercise all lie within one or the other of two different intervals.

If you need to see some worked examples, try googling keywords like: how to solve inequalities with multiple absolute values :cool:

 

[tkhunny]

Cases, not solutions.

This is all you need.

\(\displaystyle |x| = x\;if\;x \ge 0\)

\(\displaystyle |x| = -x\;if\;x<0\)