What is the solution set of the inequality |x+ 2|−|x−3| ≤ 1?

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Answer to a math question What is the solution set of the inequality |x+ 2|−|x−3| ≤ 1?

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$\begin{array} { l }\begin{array} { l }x+2-\left( x-3 \right) \leq 1,& \begin{array} { l }x+2 \geq 0,& x-3 \geq 0\end{array}\end{array},\\\begin{array} { l }-\left( x+2 \right)-\left( x-3 \right) \leq 1,& \begin{array} { l }x+2 < 0,& x-3 \geq 0\end{array}\end{array},\\\begin{array} { l }x+2-\left( -\left( x-3 \right) \right) \leq 1,& \begin{array} { l }x+2 \geq 0,& x-3 < 0\end{array}\end{array},\\\begin{array} { l }-\left( x+2 \right)-\left( -\left( x-3 \right) \right) \leq 1,& \begin{array} { l }x+2 < 0,& x-3 < 0\end{array}\end{array}\end{array}$

$\begin{array} { l }\begin{array} { l }∅,& \begin{array} { l }x+2 \geq 0,& x-3 \geq 0\end{array}\end{array},\\\begin{array} { l }-\left( x+2 \right)-\left( x-3 \right) \leq 1,& \begin{array} { l }x+2 < 0,& x-3 \geq 0\end{array}\end{array},\\\begin{array} { l }x+2-\left( -\left( x-3 \right) \right) \leq 1,& \begin{array} { l }x+2 \geq 0,& x-3 < 0\end{array}\end{array},\\\begin{array} { l }-\left( x+2 \right)-\left( -\left( x-3 \right) \right) \leq 1,& \begin{array} { l }x+2 < 0,& x-3 < 0\end{array}\end{array}\end{array}$

$\begin{array} { l }\begin{array} { l }∅,& \begin{array} { l }x \geq -2,& x-3 \geq 0\end{array}\end{array},\\\begin{array} { l }-\left( x+2 \right)-\left( x-3 \right) \leq 1,& \begin{array} { l }x+2 < 0,& x-3 \geq 0\end{array}\end{array},\\\begin{array} { l }x+2-\left( -\left( x-3 \right) \right) \leq 1,& \begin{array} { l }x+2 \geq 0,& x-3 < 0\end{array}\end{array},\\\begin{array} { l }-\left( x+2 \right)-\left( -\left( x-3 \right) \right) \leq 1,& \begin{array} { l }x+2 < 0,& x-3 < 0\end{array}\end{array}\end{array}$

$\begin{array} { l }\begin{array} { l }∅,& \begin{array} { l }x \geq -2,& x \geq 3\end{array}\end{array},\\\begin{array} { l }-\left( x+2 \right)-\left( x-3 \right) \leq 1,& \begin{array} { l }x+2 < 0,& x-3 \geq 0\end{array}\end{array},\\\begin{array} { l }x+2-\left( -\left( x-3 \right) \right) \leq 1,& \begin{array} { l }x+2 \geq 0,& x-3 < 0\end{array}\end{array},\\\begin{array} { l }-\left( x+2 \right)-\left( -\left( x-3 \right) \right) \leq 1,& \begin{array} { l }x+2 < 0,& x-3 < 0\end{array}\end{array}\end{array}$

$\begin{array} { l }\begin{array} { l }∅,& \begin{array} { l }x \geq -2,& x \geq 3\end{array}\end{array},\\\begin{array} { l }x \geq 0,& \begin{array} { l }x+2 < 0,& x-3 \geq 0\end{array}\end{array},\\\begin{array} { l }x+2-\left( -\left( x-3 \right) \right) \leq 1,& \begin{array} { l }x+2 \geq 0,& x-3 < 0\end{array}\end{array},\\\begin{array} { l }-\left( x+2 \right)-\left( -\left( x-3 \right) \right) \leq 1,& \begin{array} { l }x+2 < 0,& x-3 < 0\end{array}\end{array}\end{array}$

$\begin{array} { l }\begin{array} { l }∅,& \begin{array} { l }x \geq -2,& x \geq 3\end{array}\end{array},\\\begin{array} { l }x \geq 0,& \begin{array} { l }x < -2,& x-3 \geq 0\end{array}\end{array},\\\begin{array} { l }x+2-\left( -\left( x-3 \right) \right) \leq 1,& \begin{array} { l }x+2 \geq 0,& x-3 < 0\end{array}\end{array},\\\begin{array} { l }-\left( x+2 \right)-\left( -\left( x-3 \right) \right) \leq 1,& \begin{array} { l }x+2 < 0,& x-3 < 0\end{array}\end{array}\end{array}$

$\begin{array} { l }\begin{array} { l }∅,& \begin{array} { l }x \geq -2,& x \geq 3\end{array}\end{array},\\\begin{array} { l }x \geq 0,& \begin{array} { l }x < -2,& x \geq 3\end{array}\end{array},\\\begin{array} { l }x+2-\left( -\left( x-3 \right) \right) \leq 1,& \begin{array} { l }x+2 \geq 0,& x-3 < 0\end{array}\end{array},\\\begin{array} { l }-\left( x+2 \right)-\left( -\left( x-3 \right) \right) \leq 1,& \begin{array} { l }x+2 < 0,& x-3 < 0\end{array}\end{array}\end{array}$

$\begin{array} { l }\begin{array} { l }∅,& \begin{array} { l }x \geq -2,& x \geq 3\end{array}\end{array},\\\begin{array} { l }x \geq 0,& \begin{array} { l }x < -2,& x \geq 3\end{array}\end{array},\\\begin{array} { l }x \leq 1,& \begin{array} { l }x+2 \geq 0,& x-3 < 0\end{array}\end{array},\\\begin{array} { l }-\left( x+2 \right)-\left( -\left( x-3 \right) \right) \leq 1,& \begin{array} { l }x+2 < 0,& x-3 < 0\end{array}\end{array}\end{array}$

$\begin{array} { l }\begin{array} { l }∅,& \begin{array} { l }x \geq -2,& x \geq 3\end{array}\end{array},\\\begin{array} { l }x \geq 0,& \begin{array} { l }x < -2,& x \geq 3\end{array}\end{array},\\\begin{array} { l }x \leq 1,& \begin{array} { l }x \geq -2,& x-3 < 0\end{array}\end{array},\\\begin{array} { l }-\left( x+2 \right)-\left( -\left( x-3 \right) \right) \leq 1,& \begin{array} { l }x+2 < 0,& x-3 < 0\end{array}\end{array}\end{array}$

$\begin{array} { l }\begin{array} { l }∅,& \begin{array} { l }x \geq -2,& x \geq 3\end{array}\end{array},\\\begin{array} { l }x \geq 0,& \begin{array} { l }x < -2,& x \geq 3\end{array}\end{array},\\\begin{array} { l }x \leq 1,& \begin{array} { l }x \geq -2,& x < 3\end{array}\end{array},\\\begin{array} { l }-\left( x+2 \right)-\left( -\left( x-3 \right) \right) \leq 1,& \begin{array} { l }x+2 < 0,& x-3 < 0\end{array}\end{array}\end{array}$

$\begin{array} { l }\begin{array} { l }∅,& \begin{array} { l }x \geq -2,& x \geq 3\end{array}\end{array},\\\begin{array} { l }x \geq 0,& \begin{array} { l }x < -2,& x \geq 3\end{array}\end{array},\\\begin{array} { l }x \leq 1,& \begin{array} { l }x \geq -2,& x < 3\end{array}\end{array},\\\begin{array} { l }x \in ℝ,& \begin{array} { l }x+2 < 0,& x-3 < 0\end{array}\end{array}\end{array}$

$\begin{array} { l }\begin{array} { l }∅,& \begin{array} { l }x \geq -2,& x \geq 3\end{array}\end{array},\\\begin{array} { l }x \geq 0,& \begin{array} { l }x < -2,& x \geq 3\end{array}\end{array},\\\begin{array} { l }x \leq 1,& \begin{array} { l }x \geq -2,& x < 3\end{array}\end{array},\\\begin{array} { l }x \in ℝ,& \begin{array} { l }x < -2,& x-3 < 0\end{array}\end{array}\end{array}$

$\begin{array} { l }\begin{array} { l }∅,& \begin{array} { l }x \geq -2,& x \geq 3\end{array}\end{array},\\\begin{array} { l }x \geq 0,& \begin{array} { l }x < -2,& x \geq 3\end{array}\end{array},\\\begin{array} { l }x \leq 1,& \begin{array} { l }x \geq -2,& x < 3\end{array}\end{array},\\\begin{array} { l }x \in ℝ,& \begin{array} { l }x < -2,& x < 3\end{array}\end{array}\end{array}$

$\begin{array} { l }\begin{array} { l }∅,& x \in \left[ 3, +\infty\right\rangle\end{array},\\\begin{array} { l }x \geq 0,& \begin{array} { l }x < -2,& x \geq 3\end{array}\end{array},\\\begin{array} { l }x \leq 1,& \begin{array} { l }x \geq -2,& x < 3\end{array}\end{array},\\\begin{array} { l }x \in ℝ,& \begin{array} { l }x < -2,& x < 3\end{array}\end{array}\end{array}$

$\begin{array} { l }\begin{array} { l }∅,& x \in \left[ 3, +\infty\right\rangle\end{array},\\\begin{array} { l }x \geq 0,& ∅\end{array},\\\begin{array} { l }x \leq 1,& \begin{array} { l }x \geq -2,& x < 3\end{array}\end{array},\\\begin{array} { l }x \in ℝ,& \begin{array} { l }x < -2,& x < 3\end{array}\end{array}\end{array}$

$\begin{array} { l }\begin{array} { l }∅,& x \in \left[ 3, +\infty\right\rangle\end{array},\\\begin{array} { l }x \geq 0,& ∅\end{array},\\\begin{array} { l }x \leq 1,& x \in \left[ -2, 3\right\rangle\end{array},\\\begin{array} { l }x \in ℝ,& \begin{array} { l }x < -2,& x < 3\end{array}\end{array}\end{array}$

$\begin{array} { l }\begin{array} { l }∅,& x \in \left[ 3, +\infty\right\rangle\end{array},\\\begin{array} { l }x \geq 0,& ∅\end{array},\\\begin{array} { l }x \leq 1,& x \in \left[ -2, 3\right\rangle\end{array},\\\begin{array} { l }x \in ℝ,& x \in \langle-\infty, -2\rangle\end{array}\end{array}$

$\begin{array} { l }∅,\\\begin{array} { l }x \geq 0,& ∅\end{array},\\\begin{array} { l }x \leq 1,& x \in \left[ -2, 3\right\rangle\end{array},\\\begin{array} { l }x \in ℝ,& x \in \langle-\infty, -2\rangle\end{array}\end{array}$

$\begin{array} { l }∅,\\∅,\\\begin{array} { l }x \leq 1,& x \in \left[ -2, 3\right\rangle\end{array},\\\begin{array} { l }x \in ℝ,& x \in \langle-\infty, -2\rangle\end{array}\end{array}$

$\begin{array} { l }∅,\\∅,\\x \in \left[ -2, 1\right],\\\begin{array} { l }x \in ℝ,& x \in \langle-\infty, -2\rangle\end{array}\end{array}$

$\begin{array} { l }∅,\\∅,\\x \in \left[ -2, 1\right],\\x \in \langle-\infty, -2\rangle\end{array}$

$\begin{align*}&x \in \left\langle-\infty, 1\right] \\&\begin{array} { l }x \leq 1,& \left\{ \right\}\end{array}\end{align*}$

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