Answer :
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.
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.
