https://file.aiquickdraw.com/mathbot/file/1726197025110k0ue0lbn.jpg

The inequality provided is:

\[
4x + 5y > -15
\]

To graph this inequality, follow these steps:

### Step 1: Convert to an equation
To graph the boundary, we convert the inequality into an equation:
\[
4x + 5y = -15
\]

This is the line that will divide the plane into two regions, and the inequality will tell us which region to shade.

### Step 2: Find intercepts
To graph the line, we can find the x- and y-intercepts.

- **For the x-intercept**, set \( y = 0 \):
  \[
  4x = -15 \quad \Rightarrow \quad x = -\frac{15}{4} \approx -3.75
  \]

- **For the y-intercept**, set \( x = 0 \):
  \[
  5y = -15 \quad \Rightarrow \quad y = -3
  \]

Now you have two points: \( (-3.75, 0) \) and \( (0, -3) \).

### Step 3: Draw the boundary line
Plot these two points on the graph and draw a **dashed line** through them. The line should be dashed because the inequality is strict ( \( > \) ) and not \( \geq \).

### Step 4: Test a point
Choose a test point that is not on the line to determine which side of the line to shade. A simple point to test is \( (0, 0) \).

Substitute \( (0, 0) \) into the inequality:
\[
4(0) + 5(0) > -15 \quad \Rightarrow \quad 0 > -15
\]
This is true, so the region that contains \( (0, 0) \) is the solution region. Shade this side of the line.

### Conclusion:
- Draw a dashed line for the boundary \( 4x + 5y = -15 \).
- Shade the region that includes \( (0, 0) \).

Would you like a graphical representation of the solution? Let me know if you'd like further details or explanations.

### Related Questions:
1. How would the graph change if the inequality were \( 4x + 5y \geq -15 \)?
2. How do you find the slope of the line \( 4x + 5y = -15 \)?
3. What does it mean if the inequality sign is reversed, like \( 4x + 5y < -15 \)?
4. How would you handle a similar inequality if it were quadratic instead of linear?
5. How does the graph change if you multiply the entire inequality by a negative number?

### Tip:
When testing a point to determine the shading of an inequality, always choose a point that is easy to calculate with, such as \( (0, 0) \), unless it lies on the boundary.

Graph the following inequality: 4x + 5y > -15

Learn how to graph the linear inequality 4x + 5y > -15 step-by-step. This guide covers finding intercepts, drawing boundary lines, and shading the correct solution region. Suitable for students in Grades 8-10.

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