"Graph each system of linear inequalities. Give two ordered pairs that are solutions and two that are not solutions."
How do I find these solutions in order to "give two ordered pairs that are solutions and two that are not solutions."?
Can someone please explain this to me?
To find the solutions for the system of linear inequalities, you need to solve each inequality separately and then determine the overlapping region on the graph.
Here are the steps to find the solutions and ordered pairs for a system of linear inequalities:
1. Write down the given inequalities. For example, let's consider the following system:
- 2x + y ≥ 4
- x - y < 2
2. Solve each inequality separately for y in terms of x. For the first inequality, rearrange it to isolate y:
y ≥ -2x + 4
For the second inequality, rearrange it to isolate y:
y > x - 2
3. Graph each inequality on the same coordinate plane. Since the first inequality has a "greater than or equal to" condition, you will draw a solid line. For the second inequality, which has a "greater than" condition, you will draw a dashed line to represent the exclusion. Shade the region above the first line and below the second line.
4. Find two ordered pairs within the shaded region. To do this, pick any point within the overlapping region and verify if it satisfies both inequalities. For example, let's choose (0, 1) and (2, 5). Substitute these values into each inequality and check if they are true for both.
- For (0, 1):
Substitute x = 0 and y = 1 into the first inequality: 2(0) + 1 = 1 ≥ 4 (False)
Substitute x = 0 and y = 1 into the second inequality: 0 - 1 = -1 < 2 (True)
- For (2, 5):
Substitute x = 2 and y = 5 into the first inequality: 2(2) + 5 = 9 ≥ 4 (True)
Substitute x = 2 and y = 5 into the second inequality: 2 - 5 = -3 < 2 (True)
Therefore, (0, 1) is a solution to the system, while (2, 5) is not.
By following these steps, you can graph the system of linear inequalities and find the desired ordered pairs that are solutions and those that are not solutions.
To find solutions to a system of linear inequalities, you need to follow these steps:
1. Graph each inequality separately: Start by graphing each inequality in the system on the same coordinate plane. Treat each inequality as if it were an equation by replacing the inequality symbol with an equal sign. This will give you the boundary line for each inequality.
2. Determine the shading: Determine whether to shade the region above or below each boundary line, usually by testing a point within the region. For example, if the original inequality is ">," test a point above the line. If the point satisfies the inequality, shade the region above the line. If it does not satisfy the inequality, shade the region below the line.
3. Identify the overlapping region: Look for the region where the shaded areas of the individual inequalities overlap. This region represents the solution to the system of inequalities.
4. Find solution ordered pairs: Once you have identified the overlapping region, you can find solution ordered pairs by selecting points within that region.
To give two ordered pairs that are solutions, choose any two points within the overlapping shaded region. To give two ordered pairs that are not solutions, choose any two points outside the overlapping region.
Remember to label the graph and indicate which points are solutions and which are not based on whether they satisfy the original inequalities or not.