To solve the systems of inequalities and select the correct graph, follow these steps:
1. Rewrite the inequalities in slope-intercept form ( \( y = mx + b \) ):
For the first inequality \( 2x - y > 4 \):
[tex]\[
2x - y > 4 \implies -y > -2x + 4 \implies y < 2x - 4
\][/tex]
So, the inequality becomes:
[tex]\[
y < 2x - 4
\][/tex]
For the second inequality \( x + y < -1 \):
[tex]\[
x + y < -1 \implies y < -x - 1
\][/tex]
So, the inequality becomes:
[tex]\[
y < -x - 1
\][/tex]
2. Graph the boundary lines:
- For \( y = 2x - 4 \):
- This line has a slope of 2 and a y-intercept of -4.
- Plot the point (0, -4) on the y-axis.
- Use the slope (rise over run = 2/1), from (0, -4) move up 2 units and right 1 unit to plot another point (1, -2).
- Draw a dashed line through these points since the inequality is \( < \), not \( \leq \).
- For \( y = -x - 1 \):
- This line has a slope of -1 and a y-intercept of -1.
- Plot the point (0, -1) on the y-axis.
- Use the slope (rise over run = -1/1), from (0, -1) move down 1 unit and right 1 unit to plot another point (1, -2).
- Draw a dashed line through these points since the inequality is \( < \), not \( \leq \).
3. Shade the appropriate regions:
- For \( y < 2x - 4 \), shade below the line \( y = 2x - 4 \).
- For \( y < -x - 1 \), shade below the line \( y = -x - 1 \).
4. Identify the intersection of the shaded regions:
- The solution to the system of inequalities is the region where the shaded areas of both inequalities overlap.
- This region where they overlap represents the solution set for the system of inequalities.
5. Select the correct graph:
- Look for a graph where:
- There's a region below the line \( y = 2x - 4 \) and below the line \( y = -x - 1 \).
- The correct graph will have an area labeled A (for \( y < 2x - 4 \)), an area labeled B (for \( y < -x - 1 \)), and an overlapping region labeled AB.
By following these steps, you can visually inspect the graphs and select the one that accurately represents the solution set.
