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?