Question 1208939
<pre>
{{{abs(3x - abs(2x + 1)^"") = 4}}}

Four potential cases, putting all possible combinations of signs
before the expressions in parentheses that are between absolute 
value bars: 

{{{matrix(12,1,

""+(3x + (2x + 1)^"") = 4,
3x+2x+1=4,
5x=3,
x=3/5,
"checking:",
abs(3(3/5) - abs(2(3/5) + 1)^"") = 4,
abs(9/5-abs(6/5+1))=4,
abs(9/5-abs(11/5))=4,
abs(9/5-11/5)=4,
abs(-2/5)=4,
2/5=4,
matrix(1,2, "doesn't",check!))}}}       {{{matrix(12,1,

""+(3x - (2x + 1)^"") = 4,
3x-2x-1=4,
x-1=4,
x=5,
"checking:",
abs(3(5) - abs(2(5) + 1)^"") = 4,
abs(15-abs(10+1))=4,
abs(15-abs(11))=4,
abs(15-11)=4,
abs(4)=4,
4=4,
matrix(1,2, red(that),red(checks!)))}}}        {{{matrix(14,1,

""-(3x + (2x + 1)^"") = 4,
-(3x+2x+1)=4,
-(5x+1)=4,
5x+1=-4,
5x=-5,
x=-1,
"checking:",
abs(3(-1) - abs(2(-1) + 1)^"") = 4,
abs(-3-abs(-2+1))=4,
abs(-3-abs(-1))=4,
abs(-3-1)=4,
abs(-4)=4,
4=4,
matrix(1,2, red(that),red(checks!)))}}}       {{{matrix(14,1,

""-(3x - (2x + 1)^"") = 4,
-(3x-2x-1)=4,
-(x-1)=4,
-x+1=4,
-x=3,
x=-3,
"checking:",
abs(3(-3) - abs(2(-3) + 1)^"") = 4,
abs(-9-abs(-6+1))=4,
abs(-9-abs(-5))=4,
abs(-9-5)=4,
abs(-14)=4,
14=4,
matrix(1,2, "doesn't",check!))}}}

Only the second and third cases turned out to be solutions!  

Solutions: x = 5 and x = -1

Edwin</pre>