Question

1. solve and graph the given absolute value inequality. (5pts)

9|m - 8| - 10 < 26

Explanation:

Step1: Isolate the absolute value

Add 10 to both sides of the inequality \(9|m - 8| - 10 < 26\).
\(9|m - 8| - 10 + 10 < 26 + 10\)
\(9|m - 8| < 36\)

Step2: Divide by the coefficient of the absolute value

Divide both sides by 9.
\(\frac{9|m - 8|}{9} < \frac{36}{9}\)
\(|m - 8| < 4\)

Step3: Solve the compound inequality

Recall that \(|x| < a\) (where \(a>0\)) is equivalent to \(-a < x < a\). So for \(|m - 8| < 4\), we have:
\(-4 < m - 8 < 4\)

Step4: Solve for m

Add 8 to all parts of the compound inequality.
\(-4 + 8 < m - 8 + 8 < 4 + 8\)
\(4 < m < 12\)

Answer:

The solution to the inequality \(9|m - 8| - 10 < 26\) is \(4 < m < 12\) (or in interval notation, \((4, 12)\)). To graph this, we draw a number line, mark open circles at 4 and 12 (since the inequality is strict, \(<\) not \(\leq\)), and shade the region between 4 and 12.