What is the solution set of x+2≥6 and 3x≥6?

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

\[
\boxed{[4, \infty)}
\]