How do I solve an equation that has multiple absolute values?

Question

I have this humungous equation:

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

I have tried to use the snake method/interval method to solve this problem, but I failed to get all of the solutions. This is what I have tried:

https://mega.nz/#!SqBGzAxA!e7zwPu46IhqcCam04w6Aw4nObgPJcx0FAHLjNGxS4KE (I don't have enough reputation to post direct links to images of my work, so this is a shortened link)

Anyone know how to solve such a large equation?

Answer 1

I would consider points of interest to be the boundary points. $$ x= -3,-1,1,2$$

Partition the real line like $$(-\infty,-3)\cup (-3,-1)\cup (-1,1)\cup (1,2)\cup (2,\infty)$$

For the boundary points $x=-1$ is a solution. For the interior points you need to get rid of ab solute values and solve. Check for consistency and pick solutions if any.

Answer 2

Do cases.

Case 1: $x+3 < 0$

Case 2: $x+ 3 \ge 0$ but $x+1 < 0$.

Case 3: $x+1 \ge 0$ but $x-1 < 0$.

Case 4: $x-1 \ge 0$ but $x -2 < 0$.

Case 5: $x-2 \ge 0$.

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