Question

solve for z and graph the solution. -4|z + 1| > -8 click two endpoints to graph a line segment, an endpoint and an arrowhead ray, or two arrowheads to graph a line. to change endpoints from filled - in circles circles, click on them.

Explanation:

Step1: Divide both sides by -4 (reverse inequality)

When dividing an inequality by a negative number, the inequality sign flips. So we have:
$$\frac{-4|z + 1|}{-4} < \frac{-8}{-4}$$
Simplifying, we get:
$$|z + 1| < 2$$

Step2: Solve the absolute value inequality

The absolute value inequality \(|a| < b\) (where \(b>0\)) is equivalent to \(-b < a < b\). So for \(|z + 1| < 2\), we have:
$$-2 < z + 1 < 2$$

Step3: Subtract 1 from all parts

Subtract 1 from each part of the compound inequality:
$$-2 - 1 < z + 1 - 1 < 2 - 1$$
Simplifying each part:
$$-3 < z < 1$$

Answer:

The solution for \(z\) is \(-3 < z < 1\). To graph this, we use an open circle at \(-3\) and an open circle at \(1\) (since the inequality is strict, not including the endpoints) and draw a line segment between them.