Question

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solve for z and graph the solution. 3z - 1 > -4 or 3z + 8 ≤ -4. plot the endpoints. select an endpoint to change it from closed to open. select the middle of a segment, ray, or line to delete it.

Explanation:

Step1: Solve the first - inequality

Solve \(3z - 1>-4\). Add 1 to both sides: \(3z-1 + 1>-4 + 1\), which simplifies to \(3z>-3\). Divide both sides by 3: \(z>-1\).

Step2: Solve the second - inequality

Solve \(3z + 8\leq - 4\). Subtract 8 from both sides: \(3z+8 - 8\leq - 4 - 8\), which simplifies to \(3z\leq - 12\). Divide both sides by 3: \(z\leq - 4\).

Step3: Analyze the compound - inequality

The compound inequality is \(z\leq - 4\) or \(z>-1\).

Step4: Plot on the number - line

For \(z\leq - 4\), we have a closed - circle at \(z = - 4\) and a ray going to the left. For \(z>-1\), we have an open - circle at \(z=-1\) and a ray going to the right.

Answer:

The solution set is \(z\leq - 4\) or \(z>-1\). On the number - line, we have a closed - circle at \(z = - 4\) with a ray extending to the left and an open - circle at \(z=-1\) with a ray extending to the right.