To graph the inequalities in this system, we’ll first analyze each inequality separately:

1. First inequality: $y < \frac{1}{2}x + 1$

   • This inequality represents a line with a slope of $\frac{1}{2}$ and a y-intercept at $(0, 1)$.
   • Since the inequality is $y < \frac{1}{2}x + 1$, we will use a dashed line to indicate that points on the line itself are not included.
   • We shade the region below this line because of the "<" sign.

2. Second inequality: $y > -x - 2$

   • This inequality represents a line with a slope of $-1$ and a y-intercept at $(0, -2)$.
   • Since the inequality is $y > -x - 2$, we use a dashed line here as well.
   • We shade the region above this line because of the ">" sign.

Solution Steps

1. Plot the lines $y = \frac{1}{2}x + 1$ and $y = -x - 2$ using dashed lines.
2. Shade the region below the first line (representing $y < \frac{1}{2}x + 1$).
3. Shade the region above the second line (representing $y > -x - 2$).
4. The solution to the system is the area where the shaded regions overlap.

Would you like a step-by-step graphing explanation or have any questions?

Related Questions

1. What is the solution set for a system of linear inequalities?
2. How do you determine which side of the line to shade for a given inequality?
3. Why are dashed lines used instead of solid lines for strict inequalities?
4. How can you verify a point is in the solution region?
5. What are real-world applications of systems of linear inequalities?

Tip: Always check if an inequality includes an "equal to" condition to know whether to use a dashed or solid line on the graph.