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To find the solution set of the given inequalities \( x + 2 \geq 6 \) and \( 3x \geq 6 \), we will solve each inequality step by step.
### Inequality 1: \( x + 2 \geq 6 \)
1. Subtract 2 from both sides:
\[
x \geq 6 - 2
\]
\[
x \geq 4
\]
### Inequality 2: \( 3x \geq 6 \)
1. Divide both sides by 3:
\[
x \geq \frac{6}{3}
\]
\[
x \geq 2
\]
### Combined Solution Set
Now, we need to find the overlapping solution for both inequalities:
1. The first inequality gives us \( x \geq 4 \).
2. The second inequality gives us \( x \geq 2 \).
The more restrictive condition is \( x \geq 4 \).
### Conclusion
The solution set is:
\[
\{ x \mid x \geq 4 \}
\]
In interval notation, this can be represented as:
\[
[4, \infty)
\]