Question

graph the solution to the inequality on the number line.
$$ | - 6 - 3 w | \leq 9 $$

Explanation:

Step1: Apply the absolute - value inequality rule

If \(|a|\leq b\) (\(b\geq0\)), then \(-b\leq a\leq b\). Here \(a=-6 - 3w\) and \(b = 9\), so \(-9\leq-6 - 3w\leq9\).

Step2: Solve the left - hand side of the compound inequality

Add \(6\) to all parts: \(-9+6\leq-6 - 3w+6\leq9 + 6\), which simplifies to \(-3\leq-3w\leq15\).

Step3: Divide all parts by \(-3\) (and reverse the inequality signs)

\(\frac{-3}{-3}\geq\frac{-3w}{-3}\geq\frac{15}{-3}\), so \(1\geq w\geq - 5\) or \(-5\leq w\leq1\).

Answer:

On the number line, draw a solid circle at \(-5\) (because \(w=-5\) is included in the solution set, since the inequality is \(\leq\)) and a solid circle at \(1\) (because \(w = 1\) is included in the solution set). Then shade the region between \(-5\) and \(1\).