Question

solve for z and graph the solution. 4 ≥ |z - 3| 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: Recall absolute - value inequality rule

For $|a|\geq b$ ($b\geq0$), it is equivalent to $a\geq b$ or $a\leq - b$. Here $a = z - 3$ and $b = 4$, so we have $z-3\geq4$ or $z - 3\leq-4$.

Step2: Solve the first inequality $z-3\geq4$

Add 3 to both sides of the inequality: $z-3 + 3\geq4 + 3$, which simplifies to $z\geq7$.

Step3: Solve the second inequality $z - 3\leq-4$

Add 3 to both sides of the inequality: $z-3+3\leq - 4 + 3$, which simplifies to $z\leq-1$.

Answer:

The solution of the inequality $4\geq|z - 3|$ is $z\leq-1$ or $z\geq7$. On the number - line, we mark a closed circle at $z=-1$ and draw an arrow to the left, and mark a closed circle at $z = 7$ and draw an arrow to the right.