5 months ago
Q:

|3x-1|-3=x

Accepted Solution

A:

Solution variant #1.

Move the constant to the right-hand side and change its sign
$|3x-3|=x+3$
Move the variable to the left-hand side and change its sign
$|3x-3|-x=3$
Separate the equation into $2$ possible cases
$\begin{array} { l }\begin{array} { l }3x-3-x=3,& 3x-3 \geq 0\end{array},\\\begin{array} { l }-\left( 3x-3 \right)-x=3,& 3x-3 < 0\end{array}\end{array}$
Solve the equation for $x$
$\begin{array} { l }\begin{array} { l }x=3,& 3x-3 \geq 0\end{array},\\\begin{array} { l }-\left( 3x-3 \right)-x=3,& 3x-3 < 0\end{array}\end{array}$
Solve the inequality for $x$
$\begin{array} { l }\begin{array} { l }x=3,& x \geq 1\end{array},\\\begin{array} { l }-\left( 3x-3 \right)-x=3,& 3x-3 < 0\end{array}\end{array}$
Solve the equation for $x$
$\begin{array} { l }\begin{array} { l }x=3,& x \geq 1\end{array},\\\begin{array} { l }x=0,& 3x-3 < 0\end{array}\end{array}$
Solve the inequality for $x$
$\begin{array} { l }\begin{array} { l }x=3,& x \geq 1\end{array},\\\begin{array} { l }x=0,& x < 1\end{array}\end{array}$
Find the intersection
$\begin{array} { l }x=3,\\\begin{array} { l }x=0,& x < 1\end{array}\end{array}$
Find the intersection
$\begin{array} { l }x=3,\\x=0\end{array}$
The equation has $2$ solutions, so we'll label them as $x_1$ and $x_2$
$\begin{array} { l }x_1=0,& x_2=3\end{array}$