Question

solve the absolute value inequality.\\(|3x - 9| > 18\\)\\(\\)\\(\\)select the correct choice below and, if necessary, fill in the answer box to complete your choice.\\(\\)\\(\bigcirc\\) a. the solution set in interval notation is \\(\square\\).\\(\\)\\(\quad\\) (simplify your answer.)\\(\\)\\(\bigcirc\\) b. the solution set is \\(\varnothing\\).

Explanation:

Step1: Apply absolute value inequality rule

For \(|A| > B\) (where \(B>0\)), it is equivalent to \(A > B\) or \(A < -B\). So for \(|3x - 9| > 18\), we get two inequalities:
\(3x - 9 > 18\) or \(3x - 9 < - 18\)

Step2: Solve \(3x - 9 > 18\)

Add 9 to both sides:
\(3x - 9 + 9 > 18 + 9\)
\(3x > 27\)
Divide both sides by 3:
\(x > \frac{27}{3}\)
\(x > 9\)

Step3: Solve \(3x - 9 < - 18\)

Add 9 to both sides:
\(3x - 9 + 9 < - 18 + 9\)
\(3x < - 9\)
Divide both sides by 3:
\(x < \frac{-9}{3}\)
\(x < - 3\)

Step4: Write the solution in interval notation

The solutions are \(x < - 3\) or \(x > 9\). In interval notation, this is \((-\infty, - 3) \cup (9, \infty)\)

Answer:

A. The solution set in interval notation is \((-\infty, -3) \cup (9, \infty)\)