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To find the solution set for the inequalities \( x + 2 \geq 6 \) and \( 3x \geq 6 \), we will solve each inequality separately.
1. **Solve \( x + 2 \geq 6 \)**:
\[
x + 2 \geq 6
\]
Subtract 2 from both sides:
\[
x \geq 4
\]
2. **Solve \( 3x \geq 6 \)**:
\[
3x \geq 6
\]
Divide both sides by 3:
\[
x \geq 2
\]
Now, we need to find the intersection of the two solution sets:
- From the first inequality, \( x \geq 4 \).
- From the second inequality, \( x \geq 2 \).
The solution set that satisfies both inequalities is determined by the more restrictive condition, which is \( x \geq 4 \).
Thus, the solution set is:
\[
\{ x \mid x \geq 4 \}
\]
This can also be written as:
\[
[4, \infty)
\]