Question

solve for z and graph the solution.
\\(-2 \leq \frac{3z + 14}{2} < 7\\)
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.
(number line with ticks at -14, -12, -10, -8, -6, -4, -2, 0, 2)

Explanation:

Step1: Split the compound inequality

We have \(-2 \leq \frac{3z + 14}{2} < 7\). We can split this into two separate inequalities:

1. \(-2 \leq \frac{3z + 14}{2}\)
2. \(\frac{3z + 14}{2} < 7\)

Step2: Solve the first inequality \(-2 \leq \frac{3z + 14}{2}\)

Multiply both sides by 2 (since 2 is positive, the inequality sign remains the same):
\(-2\times2 \leq 3z + 14\)
\(-4 \leq 3z + 14\)
Subtract 14 from both sides:
\(-4 - 14 \leq 3z\)
\(-18 \leq 3z\)
Divide both sides by 3:
\(-6 \leq z\)

Step3: Solve the second inequality \(\frac{3z + 14}{2} < 7\)

Multiply both sides by 2:
\(3z + 14 < 7\times2\)
\(3z + 14 < 14\)
Subtract 14 from both sides:
\(3z < 14 - 14\)
\(3z < 0\)
Divide both sides by 3:
\(z < 0\)

Step4: Combine the two solutions

From Step2, we have \(z \geq -6\) (note: \(-6 \leq z\) is the same as \(z \geq -6\)) and from Step3, we have \(z < 0\). So the solution to the compound inequality is \(-6 \leq z < 0\)

Answer:

The solution for \(z\) is \(-6 \leq z < 0\)