Question

solve for z and graph the solution.
1 ≥ |z + 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.
submit

Explanation:

Step1: Solve the absolute - value inequality

Given \(1\geq|z + 4|\). By the property of absolute - value \(|a|\leq b\) (where \(b\geq0\)) is equivalent to \(-b\leq a\leq b\). Here \(a = z + 4\) and \(b = 1\), so \(-1\leq z+4\leq1\).
Subtract 4 from all parts of the compound inequality:
\(-1-4\leq z+4 - 4\leq1 - 4\).
\(-5\leq z\leq - 3\).

Step2: Graph the solution

The solution \(z\in[-5,-3]\) is a line segment on the number line. The endpoints \(z=-5\) and \(z = - 3\) are included (since the original inequality is \(\geq\) for the absolute - value, which when rewritten as a compound inequality \(-1\leq z + 4\leq1\) has \(\leq\) signs, corresponding to closed - circle endpoints on the number line).

Answer:

The solution of the inequality \(1\geq|z + 4|\) is \(z\in[-5,-3]\). On the number line, we click on the points \(-5\) and \(-3\) (filled - in circles) and then click on the line segment between them.