Question

solve the following inequality. graph the solution set.
|x - 5| + 3 < 9

select the correct choice below and, if necessary, fill in the answer box to complete
your choice.
○ a. there are finitely many solutions. the solution set is □.
(use a comma to separate answers as needed.)
○ b. there are infinitely many solutions. the solution set is □.
(type your answer in interval notation.)
○ c. there is no real solution.
graph the solution set. choose the correct graph below.
○ a. ←←(-8-6-4-2 0 2 4 6 8 1012141618)→→ ○ b. ←(-8-6-4-2 0 2 4 6 8 1012141618)→
○ c. ←(-8-6-4-2 0 2 4 6 8 1012141618)→ ○ d. ←(-8-6-4-2 0 2 4 6 8 ●1012141618)→
○ e. ←(-8-6-4-2 0 2 4 6 8 1012141618→ ○ f. ←(-8-6-4-●2 0 2 4 6 8 1012141618)→
○ g. ←(-8-6-4-2 0 2 4 6 8 1012141618→ ○ h. ←(-8-6-4-2 0 2 4 6 8 1012141618)→

Explanation:

Step1: Isolate the absolute value expression

Subtract 3 from both sides of the inequality \(|x - 5|+3\lt9\).
\(|x - 5|+3 - 3\lt9 - 3\)
\(|x - 5|\lt6\)

Step2: Solve the compound inequality

Recall that \(|a|\lt b\) (where \(b\gt0\)) is equivalent to \(-b\lt a\lt b\). So for \(|x - 5|\lt6\), we have:
\(-6\lt x - 5\lt6\)

Step3: Solve for x

Add 5 to all parts of the compound inequality:
\(-6 + 5\lt x - 5+5\lt6 + 5\)
\(-1\lt x\lt11\)

Answer:

B. There are infinitely many solutions. The solution set is \((-1, 11)\).

For the graph, the correct graph is option C (the one with a blue line between - 1 and 11, open circles at - 1 and 11) as it represents the interval \((-1,11)\) where all real numbers between - 1 and 11 (not including - 1 and 11) are solutions.