Question

solve for z and graph the solution.
1 ≤ |z - 6|
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: Use the property of absolute value inequality

For \(|a|\geq b\) (\(b\geq0\)), it is equivalent to \(a\geq b\) or \(a\leq -b\).
Here \(a = z - 6\) and \(b = 1\), so we have \(z-6\geq1\) or \(z - 6\leq-1\).

Step2: Solve the first inequality \(z-6\geq1\)

Add \(6\) to both sides of the inequality \(z-6\geq1\).
\(z-6 + 6\geq1+6\), which gives \(z\geq7\).

Step3: Solve the second inequality \(z - 6\leq-1\)

Add \(6\) to both sides of the inequality \(z - 6\leq-1\).
\(z-6+6\leq-1 + 6\), which gives \(z\leq5\).

Answer:

The solution of the inequality \(1\leq|z - 6|\) is \(z\leq5\) or \(z\geq7\). On the number - line, we graph a ray starting from \(z = 5\) (filled - in circle) and going to the left (towards \(-\infty\)) and a ray starting from \(z = 7\) (filled - in circle) and going to the right (towards \(+\infty\)).