the problem is...

i wasn't sure how to solve this problem. after looking in the back of the book (which only shows final answers, not the steps to get there) i determined that they solved by setting |x-7| to -(x-7) and +(x-7) and, in both instances, setting |x-1| to (x-1).

Why did they not set |x-1| to (-x+1) and (x-1) as well? this is how i approached the problem, which means i worked out four different equations... ++, +-, -+, --

this is not the correct way to approach the problem, as it results in "no solution."

I see how they arrived at the solution, i just don't understand why.

note: oddly, when i changed 3|x-7| to 3x-21 such that..

and worked the two equations...

and

i ended up with a completely different solution set.

when there is an absolute value sign on both sides of the inequality, how does one know which to "leave alone" aka take the positive value of, and which to substitute in the negative and positive values.