you just take away the positive value sign and solve for x and you get 1.
the expression inside the positive value sign can be negative.
you would get -(x + 1) - (3x+4) - (2x+1) - (x+1) = 12
that becomes:
- 3x - 4 - 2x - 1 - x - 1 = 12
combine like terms to get: -6x - 6 = 12
add 6 to both sides to get -6x = 18
solve for x to get x = -3
so, if i did this right, x = 1 or x = -3.
you can also solve it graphically.
your graph would look like this.
abs(x) means the same thing as |x| to this graphing software.
the main idea is that if the absolute value of an expression is equal to something, then either that expression is equal to that something or that expression times -1 is equal to that something.
for example:
abs(3x+4) = 7
either (3x+4) = 7 or -(3x+4) = 7
this gets you x = 1 or x = -11/3
the books will usually show you that:
abs(x) = x and abs (x) = -x
what they are talking about is the expression inside the absolute value sign, an not just the value of x.
therefore:
abs(x + a) = x + a and abs(x) = -(x + a).
i used that fact to solve this problem.
the expression inside the absolute value sign will either be positive or it will be negative.
you will also see:
if abs(x+a) > y, then (x+a) > y and (x+a) < -y.
this also stems from what i just told you.
in my interpretation:
(x+a) > y and -(x+a) > y
in the second inequality, just multiply both sides of the inequality by -1 and you will get (x+a) < -y
the books have done that extra step for you but do not usually explain why it is that way.