Question: Solve: \( x \leq -3 \) or \( 2x > 0 \) Graph the solutions on a number line. -8 -7 -6 -5 -4…

Solve: \( x \leq -3 \) or \( 2x > 0 \)

Graph the solutions on a number line.

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

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Solution

To solve the inequality \( x \leq -3 \) or \( 2x > 0 \), proceed with the following steps: First, solve \( x \leq -3 \). This inequality is already solved for \( x \). Next, solve \( 2x > 0 \). Divide both sides by 2: \[ x > 0 \] Now combine the solutions using the word “or”: \[ \begin{cases} x \leq -3 & \\ x > 0 & \end{cases} \] Graph the solutions on a number line by shading the regions that satisfy either inequality: - Shade all numbers less than or equal to \(-3\) with a closed circle at \(-3\). - Shade all numbers greater than 0 with an open circle at 0. The graph will show the union of these two intervals on the number line.