Question 825006
2x+1>0:
2x+1=x+c
'
2x+1<0:
-2x-1=x+c


Use the two resulting equations.
2x+1-x=c and -2x-1-x=c
x+1=c and -3x-1=c
Giving {{{x+1=-3x-1}}}
{{{4x=-2}}}
{{{x=-1/2}}}
That means:  {{{c=1-1/2=1/2}}} and {{{c=3(1/2)-1=3/2-2/2=1/2}}}


The equation would have just one value for c.
{{{c=1/2}}}


What if you would try solving for c first instead of x?
x+1=c and -3x-1=c
x=c-1 and -3x=c+1
x=c-1 and x=(c+1)/(-3)
then
c-1=-(c+1)/3
3c-3=-c-1
4c=-1+3
4c=2
{{{c=1/2}}}, just like before.