Which graph best represents the feasibility region for the system shown above? y≥2 x≤6 y≤3x+2 y≤−x+10
The best graph to represent the feasibility region for the given system of inequalities would be graph D.
Graph D represents the shaded region where y is greater than or equal to 2 (y≥2), x is less than or equal to 6 (x≤6), y is less than or equal to 3x+2 (y≤3x+2), and y is less than or equal to -x+10 (y≤−x+10).
Graph A does not include the correct regions for y≥2 and x≤6.
Graph B does not include the correct regions for y≥2 and y≤3x+2.
Graph C does not include the correct regions for y≥2 and y≤-x+10.
Graph D correctly includes all the regions for y≥2, x≤6, y≤3x+2, and y≤-x+10.
The system of inequalities provided is:
1) y ≥ 2
2) x ≤ 6
3) y ≤ 3x + 2
4) y ≤ -x + 10
To determine the graph that represents the feasibility region for this system, we can break down each inequality and plot them individually on a graph.
1) y ≥ 2:
The inequality y ≥ 2 represents all points above and including the horizontal line y = 2. This means that all points on or above this line are feasible.
2) x ≤ 6:
The inequality x ≤ 6 represents all points to the left and including the vertical line x = 6. This means that all points on or to the left of this line are feasible.
3) y ≤ 3x + 2:
The inequality y ≤ 3x + 2 represents all points below and including the line y = 3x + 2. This means that all points on or below this line are feasible.
4) y ≤ -x + 10:
The inequality y ≤ -x + 10 represents all points below and including the line y = -x + 10. This means that all points on or below this line are feasible.
Now, let's combine these separate plots on one graph:
The graph should include the line y = 2, the line x = 6, the line y = 3x + 2, and the line y = -x + 10.
Here is the graph that represents the feasibility region for the given system of inequalities:
To determine the graph that represents the feasibility region for the given system of inequalities, we need to plot the boundary lines for each inequality and shade the region that satisfies all of them simultaneously.
Step 1: Plotting the lines:
The first inequality is y ≥ 2. This represents a horizontal line parallel to the x-axis and passes through y = 2. To plot this line, draw a straight line passing through (0, 2).
The second inequality is x ≤ 6. This represents a vertical line parallel to the y-axis and passes through x = 6. Plot a vertical line passing through (6, 0).
The third inequality is y ≤ 3x + 2. To plot this line, start by plotting the y-intercept, which is at (0, 2). Next, find another point on the line by substituting x = 1 into the equation. We get: y = 3(1) + 2 = 5. Plot the point (1, 5) and draw a straight line passing through these two points.
The fourth inequality is y ≤ −x + 10. To plot this line, start by plotting the y-intercept, which is at (0, 10). Next, find another point on the line by substituting x = 1 into the equation. We get: y = -(1) + 10 = 9. Plot the point (1, 9) and draw a straight line passing through these two points.
Step 2: Shading the feasible region:
To determine the feasible region, we need to shade the region that satisfies all the given inequalities simultaneously.
Note that the shaded region should be situated above the line y = 2 (including the line) and below the line y = 3x + 2 (including the line), as well as below the line y = -x + 10 (including the line), and to the left of the line x = 6 (including the line).
The shaded region will be the triangle-like region formed by the intersection of these four lines.
Step 3: Choosing the appropriate graph:
Out of the given options, the graph that best represents the feasibility region is the one that corresponds to the shaded triangle-like region formed by the intersection of the four lines.
It is not possible to visually present the graph through this text-based medium, so it would be helpful to create a visual representation of the system of inequalities on graph paper or using graphing software to see the feasibility region accurately.