Answer:
x = 1/10 or x = 5/8
Explanation:
You want the solutions to the absolute value equation ...
6|3x -1| +4 = 2|x +8| -8
Simplify
Subtracting the right side expression, we have ...
6|3x -1| +4 -2|x +8| +8 = 0
3|3x -1| -|x +8| +6 = 0 . . . . . . . divide by 2
Domain
The two absolute value functions divide the domain of the expression into three parts. The boundaries of these domains are ...
3x -1 = 0 ⇒ x = 1/3
x +8 = 0 ⇒ x = -8
x ≥ 1/3
In this domain, both absolute value expressions do nothing, as their arguments are both positive.
3(3x -1) -(x +8) +6 = 0
9x -3 -x -8 +6 = 0
8x -5 = 0
x = 5/8 . . . . . one solution to the equation
-8 ≤ x ≤ 1/3
In this domain, the left absolute value expression has a negative argument, so the equation becomes ...
-3(3x -1) -(x +8) +6 = 0
-9x +3 -x -8 +6 = 0
-10x +1 = 0
x = 1/10 . . . . . . another solution to the equation
x < -8
In this domain, both absolute value expressions have negative arguments, so the equation becomes ...
-3(3x -1) +(x +8) +6 = 0
-9x +3 +x +8 +6 = 0
-8x +17 = 0
x = 17/8 . . . . . . . . not in the domain, not a solution
The solutions are x = 1/10 or x = 5/8.
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Additional comment
A graphing calculator often finds zeros easily, so we find it convenient to write the equation in a form that evaluates to zero at the solution points. That is what we have done above and in the attachment.
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