Question

solve the following inequality. then graph the solution set. |x + 9| < 8
a the solution is an interval. the solution is
(simplify your answer. type your answer in interval notation. use integers or fractions for any numbers in the expression.)
b the solution set is one or two points. the solution set is
(type an integer or a fraction. use a comma to separate answers as needed.)
c the solution set is ∅.
choose the correct graph below.
(graphs a - f are shown with number lines and arrows.)

Explanation:

Step1: Solve the absolute value inequality

The inequality is \(|x + 9| < 8\). By the definition of absolute value, this means \(-8 < x + 9 < 8\).

Step2: Subtract 9 from all parts

Subtract 9 from each part of the compound inequality: \(-8 - 9 < x + 9 - 9 < 8 - 9\), which simplifies to \(-17 < x < -1\).

Step3: Determine the correct graph

We need a graph that shows all real numbers between -17 and -1, not including -17 and -1 (so open circles or open endpoints on the number line). Looking at the options, option C (assuming the graph for option C has open arrows between -17 and -1) should be the correct one.

Answer:

The solution in interval notation is \((-17, -1)\). The correct graph is the one representing the interval \((-17, -1)\) (e.g., option C as per the visual cues where the line is between -17 and -1 with open endpoints).