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Perform the sign analysis, determine which interval/s satisfy the inequality, and use the notation to write the final answer



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!

Perform the sign analysis, determine which interval/s satisfy the inequality, and use the notation to write the final answer.

Learn how to solve the system of inequalities involving x + 2 > 0 and -x + 8 ≥ 0 using step-by-step solutions, sign analysis, and interval notation. This is suitable for students in Grades 7-9, covering key algebraic concepts and rules of inequalities.

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