https://file.aiquickdraw.com/mathbot/file/1729584045699dl34qpua.jpg

The image shows a system of inequalities:

1. \(3x + 5y < 6\)
2. \(y \geq 2x - 4\)
3. \(x > -3\)

To graph these inequalities and identify the values of \(x\) and \(y\) that satisfy them, follow these steps:

### Step 1: Graph the Inequality \(3x + 5y < 6\)
- Rewrite as \(5y < -3x + 6\) or \(y < -\frac{3}{5}x + \frac{6}{5}\).
- This represents a line with a slope of \(-\frac{3}{5}\) and a y-intercept at \(\frac{6}{5}\).
- Since the inequality is "<", shade the area below this line.

### Step 2: Graph the Inequality \(y \geq 2x - 4\)
- This is a line with a slope of \(2\) and a y-intercept at \(-4\).
- Since the inequality is "\(\geq\)", shade the area above this line, including the line itself.

### Step 3: Graph the Inequality \(x > -3\)
- This represents a vertical line at \(x = -3\).
- Since the inequality is ">", shade the region to the right of this line.

### Step 4: Find the Feasible Region
The solution to this system of inequalities is the region where all three shaded areas overlap. To identify the specific values of \(x\) and \(y\):
- Find the intersection points of the lines.
- Determine the region where all conditions are true.

Would you like a detailed graph of this system? If so, I can describe the feasible region further or plot it for you. 

Here are some related questions for further exploration:
1. What are the exact coordinates of the intersection points for these lines?
2. How do you determine if a point lies within the solution set of a system of inequalities?
3. What happens to the solution set if one of the inequalities changes direction (e.g., \(3x + 5y \leq 6\))?
4. How do you interpret the slope of each line in terms of the rate of change between \(x\) and \(y\)?
5. What is the impact on the solution set if we change \(x > -3\) to \(x \geq -3\)?

**Tip**: When graphing inequalities, always start by plotting the line as if it were an equation and then use the inequality to determine which side of the line to shade.

Learn how to solve and graph the system of inequalities 3x + 5y < 6, y ≥ 2x - 4, and x > -3. Understand the step-by-step process to identify the feasible region where all conditions are met. This includes the graphing of linear inequalities and finding the intersection points.

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