Let's solve the compound inequality step by step:

The given compound inequality is: $x + 2 \geq 2 \quad \text{and} \quad x + 7 \leq 6$

Step 1: Solve the first inequality

$x + 2 \geq 2$ Subtract 2 from both sides: $x \geq 0$

Step 2: Solve the second inequality

$x + 7 \leq 6$ Subtract 7 from both sides: $x \leq -1$

Step 3: Combine the inequalities

We have: $x \geq 0 \quad \text{and} \quad x \leq -1$ But there's a problem: $x \geq 0$ and $x \leq -1$ is impossible. No number can satisfy both conditions at the same time. Therefore, there is no solution.

Conclusion:

Since no $x$ can satisfy both inequalities simultaneously, the solution is the empty set, meaning there are no solutions.

The correct answer is none of the given options, but it is close to option E: The solution is all real numbers, which is incorrect. Instead, the solution is no numbers satisfy the inequality.

Would you like more details or further clarification? Here are 5 related questions that might help:

1. What is a compound inequality?
2. How do you solve inequalities involving "and" or "or" conditions?
3. What are the possible solutions when inequalities have no overlap?
4. How can inequalities with no solution be represented graphically?
5. What would change in the solution if "or" was used instead of "and"?

Tip: Always check for contradictions when dealing with compound inequalities involving "and."