https://file.aiquickdraw.com/mathbot/file/1725828675394191fq52n.jpg

Let's analyze the inequality from the image: 

The inequality provided is:
\[
5x + 4y \geq -8
\]

### Steps to graph this inequality:
1. **Rearrange into slope-intercept form**: 
   We need to express \( y \) in terms of \( x \), so let's isolate \( y \):
   \[
   4y \geq -8 - 5x
   \]
   Now, divide by 4:
   \[
   y \geq \frac{-8 - 5x}{4}
   \]
   Simplifying further:
   \[
   y \geq -2 - \frac{5}{4}x
   \]
   This is now in the form \( y \geq mx + b \), where \( m = -\frac{5}{4} \) is the slope and \( b = -2 \) is the y-intercept.

2. **Graph the boundary line**:
   - The boundary is the line \( y = -2 - \frac{5}{4}x \).
   - Since the inequality includes \( \geq \), the line will be solid, indicating that points on the line are included in the solution.
   - Plot the y-intercept \( (0, -2) \), and use the slope \( -\frac{5}{4} \) to find other points. The slope means that for every 4 units you move to the right, you go down 5 units.

3. **Shading**:
   Since the inequality is \( y \geq -2 - \frac{5}{4}x \), you will shade the area **above** the line because it includes all points where \( y \) is greater than or equal to the line.

Would you like me to go into more detail on any of these steps or solve a related problem? Here are five related questions to explore further:

1. What is the significance of the slope in this inequality, and how can it affect the graph?
2. How do you determine which region to shade in an inequality graph?
3. What would happen if the inequality were \( 5x + 4y > -8 \) instead of \( \geq \)?
4. How can you verify if a particular point lies within the solution set of this inequality?
5. How would you graph this inequality if it were \( 5x + 4y \leq -8 \)?

**Tip**: Always remember to change the inequality sign when multiplying or dividing both sides of an inequality by a negative number.

Graph the inequality on the axes: 5x + 4y ≥ -8

Learn how to graph the inequality 5x + 4y ≥ -8. This problem involves transforming the inequality into slope-intercept form and graphing it with a solid line to represent the boundary. Suitable for high school students learning algebra and inequalities.

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