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Solve an expression involving the absolute value
Answer
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Graph
$| x ^ { 2 } - 5 x | < 6$
$| x ^ { 2 } - 5 x | < 6$
Solution of inequality
$- 1 < x < 2 \text{ or } 3 < x < 6$
$| x ^{ 2 } -5x | < 6$
$- 1 < x < 2 $ or $ x > 3$
$ $ Solve a solution to $ x$
$| \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 5 } \color{#FF6800}{ x } | < \color{#FF6800}{ 6 }$
$ $ Divide the interval based on the value where the inside of the absolute value is 0 $ $
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 5 } \color{#FF6800}{ x } < \color{#FF6800}{ 6 } \left ( \text{However (or only)} \color{#FF6800}{ x } < \color{#FF6800}{ 0 } \right ) \\ \color{#FF6800}{ - } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 5 } \color{#FF6800}{ x } \right ) < \color{#FF6800}{ 6 } \left ( \text{However (or only)} \color{#FF6800}{ x } \geq \color{#FF6800}{ 0 } \right )$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 5 } \color{#FF6800}{ x } < \color{#FF6800}{ 6 } \left ( \text{However (or only)} \color{#FF6800}{ x } < \color{#FF6800}{ 0 } \right ) \\ \color{#FF6800}{ - } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 5 } \color{#FF6800}{ x } \right ) < \color{#FF6800}{ 6 } \left ( \text{However (or only)} \color{#FF6800}{ x } \geq \color{#FF6800}{ 0 } \right )$
$ $ Find the solution $ $
$\color{#FF6800}{ - } \color{#FF6800}{ 1 } < \color{#FF6800}{ x } < \color{#FF6800}{ 6 } \left ( \text{However (or only)} \color{#FF6800}{ x } < \color{#FF6800}{ 0 } \right ) \\ \color{#FF6800}{ x } < \color{#FF6800}{ 2 } $ or $ \color{#FF6800}{ x } > \color{#FF6800}{ 3 } \left ( \text{However (or only)} \color{#FF6800}{ x } \geq \color{#FF6800}{ 0 } \right )$
$\color{#FF6800}{ - } \color{#FF6800}{ 1 } < \color{#FF6800}{ x } < \color{#FF6800}{ 6 } \left ( \text{However (or only)} \color{#FF6800}{ x } < \color{#FF6800}{ 0 } \right ) \\ \color{#FF6800}{ x } < \color{#FF6800}{ 2 } $ or $ \color{#FF6800}{ x } > \color{#FF6800}{ 3 } \left ( \text{However (or only)} \color{#FF6800}{ x } \geq \color{#FF6800}{ 0 } \right )$
$ $ Make sure if the value is within the interval $ $
$\color{#FF6800}{ - } \color{#FF6800}{ 1 } < \color{#FF6800}{ x } < \color{#FF6800}{ 0 } \\ \color{#FF6800}{ 0 } \leq \color{#FF6800}{ x } < \color{#FF6800}{ 2 } $ or $ \color{#FF6800}{ x } > \color{#FF6800}{ 3 }$
$\color{#FF6800}{ - } \color{#FF6800}{ 1 } < \color{#FF6800}{ x } < \color{#FF6800}{ 0 } \\ \color{#FF6800}{ 0 } \leq \color{#FF6800}{ x } < \color{#FF6800}{ 2 } $ or $ \color{#FF6800}{ x } > \color{#FF6800}{ 3 }$
$ $ Find the union of sets of each interval $ $
$\color{#FF6800}{ - } \color{#FF6800}{ 1 } < \color{#FF6800}{ x } < \color{#FF6800}{ 2 } $ or $ \color{#FF6800}{ x } > \color{#FF6800}{ 3 }$
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