To solve the inequalities \(x + 2 \geq 6\) and \(3x \geq 6\), let's solve each inequality step by step.
1. **Solve the first inequality:**
\[
x + 2 \geq 6
\]
Subtract 2 from both sides:
\[
x \geq 6 - 2
\]
\[
x \geq 4
\]
2. **Solve the second inequality:**
\[
3x \geq 6
\]
Divide both sides by 3:
\[
x \geq \frac{6}{3}
\]
\[
x \geq 2
\]
3. **Combine the solutions:**
The first inequality gives us \(x \geq 4\) and the second gives us \(x \geq 2\). To satisfy both inequalities simultaneously, we take the more restrictive condition:
\[
x \geq 4
\]
4. **Write the solution set:**
In set notation, the solution set is:
\[
\{x \mid x \geq 4\}
\]
So the final answer to your question is:
\[
\{x \mid x \geq 4\}
\]