Question

solve for r and graph the solution.
8 < |r - 1| + 4
click two endpoints to graph a line segment, an endpoint and an arrowhead to graph a ray, or
two arrowheads to graph a line. to change endpoints from filled - in circles to empty circles,
click on them.

Explanation:

Step1: Isolate the absolute - value expression

Subtract \(4\) from both sides of the inequality \(8\lt|r - 1|+4\).
We get \(8-4\lt|r - 1|+4 - 4\), which simplifies to \(4\lt|r - 1|\).
Since \(4\lt|r - 1|\) is equivalent to \(|r - 1|>4\).

Step2: Use the property of absolute - value inequalities

If \(|x|>a\) (\(a>0\)), then \(x<-a\) or \(x>a\).
For \(|r - 1|>4\), we have \(r-1<-4\) or \(r - 1>4\).

Step3: Solve the two inequalities

• Solve \(r-1<-4\):

Add \(1\) to both sides: \(r-1 + 1<-4+1\), so \(r<-3\).

• Solve \(r - 1>4\):

Add \(1\) to both sides: \(r-1+1>4 + 1\), so \(r>5\).

Answer:

The solution of the inequality is \(r<-3\) or \(r>5\). On the number - line, we have an open circle at \(r=-3\) and an arrow pointing to the left, and an open circle at \(r = 5\) and an arrow pointing to the right.