#### Answer

The solutions are $x=1$ and $x=0$

#### Work Step by Step

$|x+1|=|1-3x|$
Solving this absolute value equation is equivalent to solving two separate equations, which are:
$x+1=1-3x$ and $x+1=-(1-3x)$
$\textbf{Solve the first equation:}$
$x+1=1-3x$
Take $3x$ to the left side and $1$ to the right side:
$x+3x=1-1$
$4x=0$
Solve for $x$:
$x=\dfrac{0}{4}$
$x=0$
$\textbf{Solve the second equation:}$
$x+1=-(1-3x)$
$x+1=-1+3x$
Take $3x$ to the left side and $1$ to the right side:
$x-3x=-1-1$
$-2x=-2$
Solve for $x$:
$x=\dfrac{-2}{-2}$
$x=1$
The solutions are $x=1$ and $x=0$