https://file.aiquickdraw.com/mathbot/file/17286664749149j1lvw1h.jpg

The graph shows the equation \( y = -x - 9 \), and the questions ask for solving equations and inequalities based on this line. Let's go through each part:

### (a) \( -x - 9 = 0 \)
Solving this equation:
\[
-x - 9 = 0
\]
Add 9 to both sides:
\[
-x = 9
\]
Multiply both sides by \(-1\):
\[
x = -9
\]
So, the solution set for (a) is:
\[
x = -9
\]

### (b) \( -x - 9 < 0 \)
Solving the inequality:
\[
-x - 9 < 0
\]
Add 9 to both sides:
\[
-x < 9
\]
Multiply both sides by \(-1\) (which reverses the inequality):
\[
x > -9
\]
In interval notation, the solution set is:
\[
(-9, \infty)
\]

### (c) \( -x - 9 \geq 0 \)
Solving this inequality:
\[
-x - 9 \geq 0
\]
Add 9 to both sides:
\[
-x \geq 9
\]
Multiply both sides by \(-1\):
\[
x \leq -9
\]
In interval notation, the solution set is:
\[
(-\infty, -9]
\]

### Summary of solutions:
- (a) \( x = -9 \)
- (b) \( (-9, \infty) \)
- (c) \( (-\infty, -9] \)

Would you like further details or explanations on any part?

### Related questions:
1. How do you solve inequalities involving negative coefficients?
2. What happens to inequalities when multiplying or dividing by a negative number?
3. Can you solve more complex inequalities involving quadratic terms?
4. How do you graph solutions of inequalities on a number line?
5. What are the differences between open and closed intervals in solutions?

**Tip:** Always remember to reverse the inequality sign when multiplying or dividing by a negative number!

Use the given graph of y = -x - 9 to solve each equation and inequality. Write the solution set to each inequality in interval notation.

Learn how to solve linear equations and inequalities based on a graph of y = -x - 9. This solution covers key algebra concepts, including interval notation and properties of inequalities. Ideal for students in Grades 9-10.

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