Question

solve the following inequality and graph the solution set.\\(|x + 6| \geq 18\\)\\((type an integer or a fraction. use a comma to separate answers as needed.)\\)\\(\bigcirc\\) c the solution set is \\(\varnothing\\).\
choose the correct graph below.\
\\(\bigcirc\\) a. \\(\cdots\\) \\(\bigcirc\\) b. \\(\cdots\\) \\(\bigcirc\\) c. \\(\cdots\\) \\(\bigcirc\\) d. \\(\cdots\\) \\(\bigcirc\\) e. \\(\cdots\\) \\(\bigcirc\\) f. \\(\cdots\\)

Explanation:

Step1: Recall absolute value inequality rule

For \(|A| \geq B\) (where \(B \geq 0\)), it is equivalent to \(A \geq B\) or \(A \leq -B\). Here, \(A = x + 6\) and \(B = 18\). So we have two inequalities: \(x + 6 \geq 18\) or \(x + 6 \leq - 18\).

Step2: Solve \(x + 6 \geq 18\)

Subtract 6 from both sides: \(x + 6 - 6 \geq 18 - 6\), which simplifies to \(x \geq 12\).

Step3: Solve \(x + 6 \leq - 18\)

Subtract 6 from both sides: \(x + 6 - 6 \leq - 18 - 6\), which simplifies to \(x \leq - 24\).

Step4: Determine the correct graph

The solution set is \(x \leq - 24\) or \(x \geq 12\). Looking at the graphs:

• Graph A has a closed dot at -24 (indicating \(x \leq -24\)) and a closed dot at 12 (indicating \(x \geq 12\)) with arrows pointing left from -24 and right from 12, which matches our solution.
• Graph B has open dots, but our inequality is \(\geq\) and \(\leq\), so we need closed dots, so B is incorrect.
• Graph C has closed dots but no arrows extending, so it's incorrect.
• Graph D has only a closed dot at -24, incorrect.
• Graph E is the entire number line, incorrect.
• Graph F has no markings, incorrect.

Answer:

The solution to the inequality \(|x + 6| \geq 18\) is \(x \leq - 24\) or \(x \geq 12\). The correct graph is A.