Question

solve the absolute value inequality. other than ∅, use interval notation to express the solution set and graph the solution set on a number line.
|7x - 4|+1 < 9

Explanation:

Step1: Isolate the absolute - value expression

Subtract 1 from both sides of the inequality \(|7x - 4|+1<9\).
\(|7x - 4|<9 - 1\), so \(|7x - 4|<8\).

Step2: Rewrite as a compound inequality

If \(|u|0\)), then \(-a < u < a\). Here \(u = 7x-4\) and \(a = 8\), so \(-8<7x - 4<8\).

Step3: Solve the compound inequality

Add 4 to all parts: \(-8 + 4<7x-4 + 4<8 + 4\), which simplifies to \(-4<7x<12\).
Then divide all parts by 7: \(-\frac{4}{7}

Step4: Write in interval notation

The solution set in interval notation is \((-\frac{4}{7},\frac{12}{7})\).

Step5: Graph on a number line

Draw a number - line. Mark the points \(-\frac{4}{7}\) and \(\frac{12}{7}\). Since the inequality is strict (\(<\)), use open circles at \(-\frac{4}{7}\) and \(\frac{12}{7}\), and shade the region between them.

Answer:

C. \((-\frac{4}{7},\frac{12}{7})\) and the graph has open - circles at \(-\frac{4}{7}\) and \(\frac{12}{7}\) with shading in between.