Let's start by analyzing both inequalities step by step and perform sign analysis for them.

Step 1: Solve each inequality separately.

First inequality:

$-x + 8 \geq 0$

To solve for $x$:

1. Subtract 8 from both sides: $-x \geq -8$
2. Multiply both sides by $-1$, remembering to reverse the inequality: $x \leq 8$

So, the solution to the first inequality is $x \leq 8$.

Second inequality:

$x + 2 > 0$

To solve for $x$:

1. Subtract 2 from both sides: $x > -2$

So, the solution to the second inequality is $x > -2$.

Step 2: Combine the two inequalities.

We have the two inequalities:

• $x \leq 8$
• $x > -2$

Step 3: Interval and notation.

The solution is the intersection of these two conditions. Therefore: $-2 < x \leq 8$

Final Answer:

In interval notation, the solution is: $(-2, 8]$

Would you like further clarification or details on the sign analysis process?

Here are 5 related questions for practice:

1. How would the solution change if the inequality was $x + 2 \geq 0$?
2. What happens if we multiply both sides of an inequality by a negative number?
3. How do you express $x \geq -1$ and $x < 3$ in interval notation?
4. How can you verify the solution of a combined inequality on a number line?
5. Can you solve $2x - 5 \leq 7$ and express the solution in interval notation?

Tip: When multiplying or dividing both sides of an inequality by a negative number, always reverse the inequality sign!