Question

solve for t and graph the solution.
|t - 9| + 3 > 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

Subtract 3 from both sides of \(|t - 9|+3>4\).
\(|t - 9|+3-3>4 - 3\)
\(|t - 9|>1\)

Step2: Solve the absolute - value inequality

Recall that if \(|x|>a\) (\(a>0\)), then \(x>a\) or \(x<-a\).
For \(|t - 9|>1\), we have \(t - 9>1\) or \(t - 9<-1\).

Case1: Solve \(t - 9>1\)

Add 9 to both sides: \(t-9 + 9>1+9\), so \(t>10\).

Case2: Solve \(t - 9<-1\)

Add 9 to both sides: \(t-9 + 9<-1+9\), so \(t<8\).

Answer:

The solution of the inequality \(|t - 9|+3>4\) is \(t<8\) or \(t>10\). On the number - line, we have a ray starting from an open circle at \(t = 8\) and going to the left (towards negative infinity), and a ray starting from an open circle at \(t = 10\) and going to the right (towards positive infinity).