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To solve the inequalities \( x + 2 \geq 6 \) and \( 3x \geq 6 \), we will solve each inequality separately.
1. **Solve the first inequality:**
\[
x + 2 \geq 6
\]
Subtract 2 from both sides:
\[
x \geq 4
\]
2. **Solve the second inequality:**
\[
3x \geq 6
\]
Divide both sides by 3:
\[
x \geq 2
\]
3. **Find the solution set:**
The first inequality gives us \( x \geq 4 \), and the second gives us \( x \geq 2 \).
The solution set that satisfies both inequalities is the more restrictive condition, which is \( x \geq 4 \).
Thus, the final solution set is
\[
\boxed{[4, \infty)}
\]