https://file.aiquickdraw.com/mathbot/file/1726955999692i6kuv99g.jpg

Let's solve and graph the system of linear inequalities provided in the image.

### The system is:
1. \( 4x + 3y > -15 \)
2. \( y \leq -\frac{4}{3}x + 1 \)

#### Step 1: Rewrite each inequality for graphing

1. **Inequality 1: \( 4x + 3y > -15 \)**
   - Solve for \( y \):
     \[
     3y > -4x - 15
     \]
     \[
     y > -\frac{4}{3}x - 5
     \]
   - This inequality represents a line with slope \( -\frac{4}{3} \) and y-intercept \( -5 \). Since the inequality is strict (">"), the line should be dashed.

2. **Inequality 2: \( y \leq -\frac{4}{3}x + 1 \)**
   - This inequality is already solved for \( y \). It represents a line with slope \( -\frac{4}{3} \) and y-intercept \( 1 \), and since the inequality is \( \leq \), the line is solid.

#### Step 2: Graph the inequalities

- **Inequality 1:**
   - Graph the dashed line \( y = -\frac{4}{3}x - 5 \).
   - Shade the region above this line, because the inequality is \( y > -\frac{4}{3}x - 5 \).

- **Inequality 2:**
   - Graph the solid line \( y = -\frac{4}{3}x + 1 \).
   - Shade the region below this line, because the inequality is \( y \leq -\frac{4}{3}x + 1 \).

#### Step 3: Find a point in the solution set

- The solution to the system is the intersection of the shaded regions from both inequalities.
- A point that lies in both shaded regions would satisfy both inequalities. A quick check for possible points shows that \( (0, 0) \) is in the solution set:
  - For \( 4x + 3y > -15 \), substituting \( x = 0 \), \( y = 0 \):
    \[
    4(0) + 3(0) = 0 > -15 \quad \text{(True)}
    \]
  - For \( y \leq -\frac{4}{3}x + 1 \), substituting \( x = 0 \), \( y = 0 \):
    \[
    0 \leq 1 \quad \text{(True)}
    \]

Thus, \( (0, 0) \) is a point in the solution set.

Would you like further details or clarification?

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Here are 5 related questions to expand on the topic:
1. What is the difference between strict and non-strict inequalities when graphing?
2. How do you determine which region to shade for a given inequality?
3. Can a system of inequalities have no solutions? If so, under what conditions?
4. How do you check if a point lies in the solution set of a system of inequalities?
5. How does changing the slope of a line affect the solution set in a system of inequalities?

**Tip:** Always use a dashed line for strict inequalities and a solid line for non-strict inequalities when graphing.

Graph the solution to the following system of inequalities:

1. 4x + 3y > -15
2. y ≤ -4/3 x + 1

Then give the coordinates of one point in the solution set.

Learn how to graph and solve the system of linear inequalities 4x + 3y > -15 and y ≤ -4/3 x + 1. This solution involves graphing each inequality and finding the intersection region. Suitable for high school students in Grades 9-12.

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