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To solve the system of inequalities and graph the solution, we can approach each inequality individually, and then determine the region where the solutions of both inequalities overlap.

The system of inequalities is:
1. \( y > -2x + 5 \)
2. \( y \geq 5x - 6 \)

### Step 1: Graph the boundary lines
- For the first inequality \( y > -2x + 5 \), the boundary line is \( y = -2x + 5 \). This line has a slope of -2 and a y-intercept at (0, 5). Since the inequality is \( > \), the region above the line will be shaded, and the line itself will be dashed because points on the line are not included in the solution.
  
- For the second inequality \( y \geq 5x - 6 \), the boundary line is \( y = 5x - 6 \). This line has a slope of 5 and a y-intercept at (0, -6). Since the inequality is \( \geq \), the region above the line will be shaded, and the line itself will be solid because points on the line are included in the solution.

### Step 2: Determine the shaded region
- For each inequality, shade the region above the respective boundary line.
- The solution to the system is where the shaded regions for both inequalities overlap.

### Step 3: Graphical solution
On the graph:
- Draw the dashed line \( y = -2x + 5 \) and shade above it.
- Draw the solid line \( y = 5x - 6 \) and shade above it.

The solution will be the region where both shaded areas intersect.

Would you like me to plot this graph for you, or would you like further details on solving the system?

### Related Questions:
1. How do we determine which side of a line to shade for an inequality?
2. What is the significance of using a dashed vs. solid line when graphing inequalities?
3. How do we find the point of intersection of two lines?
4. What would happen if the inequalities were reversed (e.g., \( y \leq -2x + 5 \))?
5. How do systems of inequalities differ from systems of equations?

**Tip**: When solving systems of inequalities graphically, always start by determining the boundary lines and then carefully identify the regions to shade based on the inequality symbols.

Graph the solution to the following system of inequalities: y > -2x + 5 and y ≥ 5x - 6.

Learn how to graph the solution for the system of inequalities y > -2x + 5 and y ≥ 5x - 6. This guide walks through graphing boundary lines and identifying overlapping shaded regions. Suitable for students in Grades 9-10.

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