Question

solve for z and graph the solution.
450 ≥ |z + 550|
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: Solve the absolute - value inequality

Given \(450\geq|z + 550|\). By the property of absolute - value \(|a|\leq b\) (in our case \(b = 450\) and \(a=z + 550\)) is equivalent to \(-b\leq a\leq b\). So, \(-450\leq z+550\leq450\).
Subtract \(550\) from all parts of the compound inequality:
\(-450-550\leq z+550 - 550\leq450-550\).
\(-1000\leq z\leq - 100\).

Step2: Graph the solution

The solution \(z\in[-1000,-100]\) is a line segment on the number line. The endpoints are \(-1000\) and \(-100\). Since the inequality is \(\geq\) for the absolute - value (which is equivalent to \(\leq\) for the compound inequality after rewriting), the endpoints are included in the solution set. So, we graph a line segment with filled - in circles at \(z=-1000\) and \(z = - 100\).

Answer:

The solution of the inequality \(450\geq|z + 550|\) is \(z\in[-1000,-100]\). Graph a line segment on the number line with endpoints at \(-1000\) and \(-100\) (both endpoints are filled).