Question

solve for z and graph the solution.
|z + 5| < 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 \(|z + 5|\lt4\). By the definition of absolute - value inequality \(|x|\lt a\) (\(a\gt0\)) which is equivalent to \(-a\lt x\lt a\). Here \(x = z + 5\) and \(a = 4\), so \(-4\lt z+5\lt4\).

Step2: Isolate \(z\)

Subtract 5 from all parts of the compound inequality.
For the left - hand side: \(-4-5\lt z+5 - 5\), which gives \(-9\lt z\).
For the right - hand side: \(z+5 - 5\lt4 - 5\), which gives \(z\lt - 1\).

Answer:

The solution of the inequality \(|z + 5|\lt4\) is \(-9\lt z\lt - 1\). On the number line, we have an open circle at \(z=-9\) (since \(z\gt - 9\)) and an open circle at \(z = - 1\) (since \(z\lt - 1\)) and a line segment connecting these two open circles.