5. A system of linear inequalities is shown below:
x + y ≥ 4
y < 2x – 3
Describe the steps to graph the solution set to the system of inequalities. Be specific. (10 points)
To graph the solution set to the system of linear inequalities, follow these steps:
1. Begin by graphing the first inequality, x + y ≥ 4:
- Rewrite the inequality in slope-intercept form: y ≥ -x + 4. This means that y is greater than or equal to the line -x + 4.
- To graph the line, start by plotting the y-intercept, which is 4. Mark a point on the y-axis at (0, 4).
- Determine the slope, which is -1 (the coefficient of x). Since the slope is negative, move down 1 unit and to the right 1 unit to find the next point. Repeat this process to plot more points.
- Draw a solid line through the plotted points to represent the inequality. The line should extend infinitely in both directions.
2. Next, graph the second inequality, y < 2x – 3:
- Rewrite the inequality in slope-intercept form: y < 2x – 3. This means that y is less than the line 2x – 3.
- To graph the line, start by plotting the y-intercept, which is -3. Mark a point on the y-axis at (0, -3).
- Determine the slope, which is 2 (the coefficient of x). Since the slope is positive, move up 2 units and to the right 1 unit to find the next point. Repeat this process to plot more points.
- Draw a dashed line through the plotted points to represent the inequality. The line should extend infinitely in both directions.
3. Remember that the inequality is y < 2x – 3, so the line should be dashed to show that the points on the line are not included in the solution set.
4. The shaded region that satisfies both inequalities is the solution set of the system of inequalities.
- Shade the area above the line y ≥ -x + 4.
- Shade the area below the dashed line y < 2x – 3.
5. The overlapping shaded region represents the solution set to the system of inequalities. The points within this shaded region satisfy both inequalities simultaneously.
6. Label the shaded region with an arrow and write "Solution Set" to indicate that it represents all the possible solutions to the system of inequalities.
These steps will help you accurately graph the solution set to the given system of linear inequalities.
To graph the solution set of a system of linear inequalities, follow these steps:
1. Begin by graphing each inequality separately. Treat each inequality as if it were an equation and draw its corresponding line on the coordinate plane.
For the inequality x + y ≥ 4:
- First, rewrite the inequality as an equation by replacing the inequality sign with an equals sign: x + y = 4.
- Next, choose any x value and solve for y to find two points on the line. For example, if x = 0, then y = 4. If x = 4, then y = 0. Plot these two points on the graph and draw a straight line passing through them.
For the inequality y < 2x - 3:
- Rewrite the inequality as an equation: y = 2x - 3.
- Choose any x value and solve for y to find two points on the line. For example, if x = 0, then y = -3. If x = 2, then y = 1. Plot these two points on the graph and draw a straight line passing through them.
2. Now, shade the regions that satisfy each inequality.
For the inequality x + y ≥ 4:
- Choose a test point not on the line, such as (0,0), and substitute its coordinates into the inequality to check if it's true.
- If the test point satisfies the inequality (e.g., 0 + 0 ≥ 4 is false), shade the region containing the test point.
- If the test point doesn't satisfy the inequality (e.g., 0 + 0 ≥ 4 is false), shade the opposite region.
For the inequality y < 2x - 3:
- Again, choose a test point not on the line, such as (0,0), and substitute its coordinates into the inequality to check if it's true.
- If the test point satisfies the inequality (e.g., 0 < 2(0) - 3 is true), shade the region containing the test point.
- If the test point doesn't satisfy the inequality (e.g., 0 < 2(0) - 3 is true), shade the opposite region.
3. Finally, find the overlapping shaded region of both inequalities. This region represents the solution set to the system of inequalities.
In this case, the overlapping shaded region would be the area below the line y = 2x - 3 and above or on the line x + y = 4.