Question

graph the solution to the inequality on the number line. |9 - 6w| < 15

Explanation:

Step1: Rewrite the absolute - value inequality

An absolute - value inequality \(|a|\lt b\) (\(b > 0\)) can be rewritten as \(-b\lt a\lt b\). So, \(|9 - 6w|\lt15\) becomes \(-15\lt9 - 6w\lt15\).

Step2: Subtract 9 from all parts of the compound inequality

\(-15-9\lt9 - 6w-9\lt15 - 9\), which simplifies to \(-24\lt - 6w\lt6\).

Step3: Divide all parts by - 6 and reverse the inequality signs

When dividing an inequality by a negative number, the inequality signs are reversed. So, \(\frac{-24}{-6}>\frac{-6w}{-6}>\frac{6}{-6}\), which gives \(4 > w>-1\) or \(-1\lt w\lt4\).

Step4: Graph on the number line

On the number line, we use an open circle at \(w=-1\) and \(w = 4\) (because the inequality is strict, i.e., \(\lt\) not \(\leq\)) and draw a line segment between them.

Answer:

On the number line, place an open circle at \(-1\), an open circle at \(4\), and draw a line segment connecting them.