Question

solve for z and graph the solution. |5 - 5z| ≥ 25. 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

For \(|15 - 5z|\geq25\), we consider two cases:
Case 1: \(15−5z\geq25\)
Subtract 15 from both sides: \(-5z\geq25 - 15\), so \(-5z\geq10\). Divide both sides by - 5 and reverse the inequality sign (since dividing by a negative number), we get \(z\leq - 2\).
Case 2: \(15−5z\leq - 25\)
Subtract 15 from both sides: \(-5z\leq - 25 - 15\), so \(-5z\leq - 40\). Divide both sides by - 5 and reverse the inequality sign, we get \(z\geq8\).

Step2: Graph the solution

On the number - line, for \(z\leq - 2\), we draw a ray starting from \(z=-2\) and going to the left with a filled - in circle at \(z = - 2\). For \(z\geq8\), we draw a ray starting from \(z = 8\) and going to the right with a filled - in circle at \(z = 8\).

Answer:

The solution of the inequality is \(z\leq - 2\) or \(z\geq8\). On the number - line, there is a ray to the left of \(z=-2\) (including \(-2\)) and a ray to the right of \(z = 8\) (including \(8\)).