Question

give the solution set in both interval and graph form.
$| - 5 + x | > 1$
the solution set is $(-infty,4)cup(6,infty)$
(type your answer in interval notation.)
the solution set is $varnothing$.
choose the correct graph of the solution set below.

Explanation:

Step1: Solve the absolute - value inequality

Recall the property \(|a|>b\) implies \(a > b\) or \(a<-b\). For \(|-5 + x|>1\), we have \(x - 5>1\) or \(x - 5<-1\).

Case1: Solve \(x - 5>1\)

Add 5 to both sides of the inequality \(x-5>1\).
\(x-5 + 5>1+5\), so \(x>6\).

Case2: Solve \(x - 5<-1\)

Add 5 to both sides of the inequality \(x - 5<-1\).
\(x-5 + 5<-1+5\), so \(x<4\).

Step2: Analyze the graph

The solution set \(x<4\) or \(x>6\) in interval notation is \((-\infty,4)\cup(6,\infty)\).
For the graph, when \(x = 4\) and \(x = 6\), since the inequality is strict (\(>\)), there are open - circles at \(x = 4\) and \(x = 6\). The graph has a ray going to the left from \(x = 4\) (towards \(-\infty\)) and a ray going to the right from \(x = 6\) (towards \(\infty\)).

Answer:

A. The solution set is \((-\infty,4)\cup(6,\infty)\) and the correct graph is the one with open - circles at \(4\) and \(6\) and rays going to the left of \(4\) and to the right of \(6\) (assuming option C is the correct graph based on the description of open - circles and the direction of rays for \(x<4\) and \(x>6\)).