Evaluate: |x+1| > |3x-5|

Expression: $|x+1| > |3x-5|$

Move the expression to the left-hand side and change its sign

$|x+1|-|3x-5| > 0$

Separate the inequality into $4$ possible cases

$\begin{array} { l }\begin{array} { l }x+1-\left( 3x-5 \right) > 0,& \begin{array} { l }x+1 \geq 0,& 3x-5 \geq 0\end{array}\end{array},\\\begin{array} { l }-\left( x+1 \right)-\left( 3x-5 \right) > 0,& \begin{array} { l }x+1 < 0,& 3x-5 \geq 0\end{array}\end{array},\\\begin{array} { l }x+1-\left( -\left( 3x-5 \right) \right) > 0,& \begin{array} { l }x+1 \geq 0,& 3x-5 < 0\end{array}\end{array},\\\begin{array} { l }-\left( x+1 \right)-\left( -\left( 3x-5 \right) \right) > 0,& \begin{array} { l }x+1 < 0,& 3x-5 < 0\end{array}\end{array}\end{array}$

Solve the inequality for $x$

$\begin{array} { l }\begin{array} { l }x < 3,& \begin{array} { l }x+1 \geq 0,& 3x-5 \geq 0\end{array}\end{array},\\\begin{array} { l }-\left( x+1 \right)-\left( 3x-5 \right) > 0,& \begin{array} { l }x+1 < 0,& 3x-5 \geq 0\end{array}\end{array},\\\begin{array} { l }x+1-\left( -\left( 3x-5 \right) \right) > 0,& \begin{array} { l }x+1 \geq 0,& 3x-5 < 0\end{array}\end{array},\\\begin{array} { l }-\left( x+1 \right)-\left( -\left( 3x-5 \right) \right) > 0,& \begin{array} { l }x+1 < 0,& 3x-5 < 0\end{array}\end{array}\end{array}$

Solve the inequality for $x$

$\begin{array} { l }\begin{array} { l }x < 3,& \begin{array} { l }x \geq -1,& 3x-5 \geq 0\end{array}\end{array},\\\begin{array} { l }-\left( x+1 \right)-\left( 3x-5 \right) > 0,& \begin{array} { l }x+1 < 0,& 3x-5 \geq 0\end{array}\end{array},\\\begin{array} { l }x+1-\left( -\left( 3x-5 \right) \right) > 0,& \begin{array} { l }x+1 \geq 0,& 3x-5 < 0\end{array}\end{array},\\\begin{array} { l }-\left( x+1 \right)-\left( -\left( 3x-5 \right) \right) > 0,& \begin{array} { l }x+1 < 0,& 3x-5 < 0\end{array}\end{array}\end{array}$

Solve the inequality for $x$

$\begin{array} { l }\begin{array} { l }x < 3,& \begin{array} { l }x \geq -1,& x \geq \frac{ 5 }{ 3 }\end{array}\end{array},\\\begin{array} { l }-\left( x+1 \right)-\left( 3x-5 \right) > 0,& \begin{array} { l }x+1 < 0,& 3x-5 \geq 0\end{array}\end{array},\\\begin{array} { l }x+1-\left( -\left( 3x-5 \right) \right) > 0,& \begin{array} { l }x+1 \geq 0,& 3x-5 < 0\end{array}\end{array},\\\begin{array} { l }-\left( x+1 \right)-\left( -\left( 3x-5 \right) \right) > 0,& \begin{array} { l }x+1 < 0,& 3x-5 < 0\end{array}\end{array}\end{array}$

Solve the inequality for $x$

$\begin{array} { l }\begin{array} { l }x < 3,& \begin{array} { l }x \geq -1,& x \geq \frac{ 5 }{ 3 }\end{array}\end{array},\\\begin{array} { l }x < 1,& \begin{array} { l }x+1 < 0,& 3x-5 \geq 0\end{array}\end{array},\\\begin{array} { l }x+1-\left( -\left( 3x-5 \right) \right) > 0,& \begin{array} { l }x+1 \geq 0,& 3x-5 < 0\end{array}\end{array},\\\begin{array} { l }-\left( x+1 \right)-\left( -\left( 3x-5 \right) \right) > 0,& \begin{array} { l }x+1 < 0,& 3x-5 < 0\end{array}\end{array}\end{array}$

Solve the inequality for $x$

$\begin{array} { l }\begin{array} { l }x < 3,& \begin{array} { l }x \geq -1,& x \geq \frac{ 5 }{ 3 }\end{array}\end{array},\\\begin{array} { l }x < 1,& \begin{array} { l }x < -1,& 3x-5 \geq 0\end{array}\end{array},\\\begin{array} { l }x+1-\left( -\left( 3x-5 \right) \right) > 0,& \begin{array} { l }x+1 \geq 0,& 3x-5 < 0\end{array}\end{array},\\\begin{array} { l }-\left( x+1 \right)-\left( -\left( 3x-5 \right) \right) > 0,& \begin{array} { l }x+1 < 0,& 3x-5 < 0\end{array}\end{array}\end{array}$

Solve the inequality for $x$

$\begin{array} { l }\begin{array} { l }x < 3,& \begin{array} { l }x \geq -1,& x \geq \frac{ 5 }{ 3 }\end{array}\end{array},\\\begin{array} { l }x < 1,& \begin{array} { l }x < -1,& x \geq \frac{ 5 }{ 3 }\end{array}\end{array},\\\begin{array} { l }x+1-\left( -\left( 3x-5 \right) \right) > 0,& \begin{array} { l }x+1 \geq 0,& 3x-5 < 0\end{array}\end{array},\\\begin{array} { l }-\left( x+1 \right)-\left( -\left( 3x-5 \right) \right) > 0,& \begin{array} { l }x+1 < 0,& 3x-5 < 0\end{array}\end{array}\end{array}$

Solve the inequality for $x$

$\begin{array} { l }\begin{array} { l }x < 3,& \begin{array} { l }x \geq -1,& x \geq \frac{ 5 }{ 3 }\end{array}\end{array},\\\begin{array} { l }x < 1,& \begin{array} { l }x < -1,& x \geq \frac{ 5 }{ 3 }\end{array}\end{array},\\\begin{array} { l }x > 1,& \begin{array} { l }x+1 \geq 0,& 3x-5 < 0\end{array}\end{array},\\\begin{array} { l }-\left( x+1 \right)-\left( -\left( 3x-5 \right) \right) > 0,& \begin{array} { l }x+1 < 0,& 3x-5 < 0\end{array}\end{array}\end{array}$

Solve the inequality for $x$

$\begin{array} { l }\begin{array} { l }x < 3,& \begin{array} { l }x \geq -1,& x \geq \frac{ 5 }{ 3 }\end{array}\end{array},\\\begin{array} { l }x < 1,& \begin{array} { l }x < -1,& x \geq \frac{ 5 }{ 3 }\end{array}\end{array},\\\begin{array} { l }x > 1,& \begin{array} { l }x \geq -1,& 3x-5 < 0\end{array}\end{array},\\\begin{array} { l }-\left( x+1 \right)-\left( -\left( 3x-5 \right) \right) > 0,& \begin{array} { l }x+1 < 0,& 3x-5 < 0\end{array}\end{array}\end{array}$

Solve the inequality for $x$

$\begin{array} { l }\begin{array} { l }x < 3,& \begin{array} { l }x \geq -1,& x \geq \frac{ 5 }{ 3 }\end{array}\end{array},\\\begin{array} { l }x < 1,& \begin{array} { l }x < -1,& x \geq \frac{ 5 }{ 3 }\end{array}\end{array},\\\begin{array} { l }x > 1,& \begin{array} { l }x \geq -1,& x < \frac{ 5 }{ 3 }\end{array}\end{array},\\\begin{array} { l }-\left( x+1 \right)-\left( -\left( 3x-5 \right) \right) > 0,& \begin{array} { l }x+1 < 0,& 3x-5 < 0\end{array}\end{array}\end{array}$

Solve the inequality for $x$

$\begin{array} { l }\begin{array} { l }x < 3,& \begin{array} { l }x \geq -1,& x \geq \frac{ 5 }{ 3 }\end{array}\end{array},\\\begin{array} { l }x < 1,& \begin{array} { l }x < -1,& x \geq \frac{ 5 }{ 3 }\end{array}\end{array},\\\begin{array} { l }x > 1,& \begin{array} { l }x \geq -1,& x < \frac{ 5 }{ 3 }\end{array}\end{array},\\\begin{array} { l }x > 3,& \begin{array} { l }x+1 < 0,& 3x-5 < 0\end{array}\end{array}\end{array}$

Solve the inequality for $x$

$\begin{array} { l }\begin{array} { l }x < 3,& \begin{array} { l }x \geq -1,& x \geq \frac{ 5 }{ 3 }\end{array}\end{array},\\\begin{array} { l }x < 1,& \begin{array} { l }x < -1,& x \geq \frac{ 5 }{ 3 }\end{array}\end{array},\\\begin{array} { l }x > 1,& \begin{array} { l }x \geq -1,& x < \frac{ 5 }{ 3 }\end{array}\end{array},\\\begin{array} { l }x > 3,& \begin{array} { l }x < -1,& 3x-5 < 0\end{array}\end{array}\end{array}$

Solve the inequality for $x$

$\begin{array} { l }\begin{array} { l }x < 3,& \begin{array} { l }x \geq -1,& x \geq \frac{ 5 }{ 3 }\end{array}\end{array},\\\begin{array} { l }x < 1,& \begin{array} { l }x < -1,& x \geq \frac{ 5 }{ 3 }\end{array}\end{array},\\\begin{array} { l }x > 1,& \begin{array} { l }x \geq -1,& x < \frac{ 5 }{ 3 }\end{array}\end{array},\\\begin{array} { l }x > 3,& \begin{array} { l }x < -1,& x < \frac{ 5 }{ 3 }\end{array}\end{array}\end{array}$

Find the intersection

$\begin{array} { l }\begin{array} { l }x < 3,& x \in \left[ \frac{ 5 }{ 3 }, +\infty\right\rangle\end{array},\\\begin{array} { l }x < 1,& \begin{array} { l }x < -1,& x \geq \frac{ 5 }{ 3 }\end{array}\end{array},\\\begin{array} { l }x > 1,& \begin{array} { l }x \geq -1,& x < \frac{ 5 }{ 3 }\end{array}\end{array},\\\begin{array} { l }x > 3,& \begin{array} { l }x < -1,& x < \frac{ 5 }{ 3 }\end{array}\end{array}\end{array}$

Find the intersection

$\begin{array} { l }\begin{array} { l }x < 3,& x \in \left[ \frac{ 5 }{ 3 }, +\infty\right\rangle\end{array},\\\begin{array} { l }x < 1,& ∅\end{array},\\\begin{array} { l }x > 1,& \begin{array} { l }x \geq -1,& x < \frac{ 5 }{ 3 }\end{array}\end{array},\\\begin{array} { l }x > 3,& \begin{array} { l }x < -1,& x < \frac{ 5 }{ 3 }\end{array}\end{array}\end{array}$

Find the intersection

$\begin{array} { l }\begin{array} { l }x < 3,& x \in \left[ \frac{ 5 }{ 3 }, +\infty\right\rangle\end{array},\\\begin{array} { l }x < 1,& ∅\end{array},\\\begin{array} { l }x > 1,& x \in \left[ -1, \frac{ 5 }{ 3 }\right\rangle\end{array},\\\begin{array} { l }x > 3,& \begin{array} { l }x < -1,& x < \frac{ 5 }{ 3 }\end{array}\end{array}\end{array}$

Find the intersection

$\begin{array} { l }\begin{array} { l }x < 3,& x \in \left[ \frac{ 5 }{ 3 }, +\infty\right\rangle\end{array},\\\begin{array} { l }x < 1,& ∅\end{array},\\\begin{array} { l }x > 1,& x \in \left[ -1, \frac{ 5 }{ 3 }\right\rangle\end{array},\\\begin{array} { l }x > 3,& x \in \langle-\infty, -1\rangle\end{array}\end{array}$

Find the intersection

$\begin{array} { l }x \in \left[ \frac{ 5 }{ 3 }, 3\right\rangle,\\\begin{array} { l }x < 1,& ∅\end{array},\\\begin{array} { l }x > 1,& x \in \left[ -1, \frac{ 5 }{ 3 }\right\rangle\end{array},\\\begin{array} { l }x > 3,& x \in \langle-\infty, -1\rangle\end{array}\end{array}$

Find the intersection

$\begin{array} { l }x \in \left[ \frac{ 5 }{ 3 }, 3\right\rangle,\\∅,\\\begin{array} { l }x > 1,& x \in \left[ -1, \frac{ 5 }{ 3 }\right\rangle\end{array},\\\begin{array} { l }x > 3,& x \in \langle-\infty, -1\rangle\end{array}\end{array}$

Find the intersection

$\begin{array} { l }x \in \left[ \frac{ 5 }{ 3 }, 3\right\rangle,\\∅,\\x \in \langle1, \frac{ 5 }{ 3 }\rangle,\\\begin{array} { l }x > 3,& x \in \langle-\infty, -1\rangle\end{array}\end{array}$

Find the intersection

$\begin{array} { l }x \in \left[ \frac{ 5 }{ 3 }, 3\right\rangle,\\∅,\\x \in \langle1, \frac{ 5 }{ 3 }\rangle,\\∅\end{array}$

Find the union

$\begin{align*}&x \in \langle1, 3\rangle\end{align*}$