Which graph is the solution of the following system? y≤2x+2 and y<-x+1

To solve the system of inequalities \( y \leq 2x + 2 \) and \( y < -x + 1 \), we need to graph the solution to both inequalities and find the region that satisfies both.

1. **Graphing \( y \leq 2x + 2 \):**
   - The boundary line is \( y = 2x + 2 \). This is a straight line with a slope of 2 and a y-intercept of 2.
   - Since the inequality is \( y \leq 2x + 2 \), we shade the region **below** or **on** the line, indicating that y can be equal to or less than the line.

2. **Graphing \( y < -x + 1 \):**
   - The boundary line is \( y = -x + 1 \). This is a straight line with a slope of -1 and a y-intercept of 1.
   - Since the inequality is \( y < -x + 1 \), we shade the region **below** the line. The boundary line is not included, so we use a **dashed** line to show that the points on the line are not part of the solution.

### Intersection of the two regions:
- The solution to the system of inequalities is the region where the shaded areas overlap.
- The graph would show the overlap between the area under \( y \leq 2x + 2 \) and the area under \( y < -x + 1 \).

### Sketching the graph:
- Plot the line \( y = 2x + 2 \) and shade below it.
- Plot the line \( y = -x + 1 \) with a dashed line and shade below it.
- The region where the shaded areas overlap is the solution to the system.

If you'd like, I can sketch this graph for you to better visualize the solution. Would you like me to do that?

Learn how to graph the solutions of the linear inequalities y ≤ 2x + 2 and y < -x + 1. Understand the concepts of shading regions under and below lines to find the intersecting solution. Suitable for high school students learning about graphing linear equations and inequalities.

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