sketch the feasible region for this system of the inqualities and find the coordinates of each vertex:y greater than or equal to 1;x less than or equal to 2; -1x+2y less than or equal to 6;1x+2y greater than or equal to 2
a) y>/= 1
b) x</=2
2y</= x+6
c) y </= (1/2) x + 3
2 y >/= -x +2
d) y >/= -(1/2)x + 1
a and d hit at (0,1)
a and b hit at (2,1)
b and c hit at (2,4)
c and d hit at (-2,3)
Inside that figure
To sketch the feasible region for this system of inequalities, we need to plot the lines corresponding to each inequality and shade in the region that satisfies all of the inequalities.
1. Start by plotting the line y = 1. Since it is "greater than or equal to," the line should be solid and include the point (0, 1).
2. Next, plot the line x = 2. Since it is "less than or equal to," the line should be solid and include the point (2, 0).
3. To plot the line -x + 2y = 6, we need to find two points on the line. Set x = 0, and solve the equation to find y: -0 + 2y = 6? y = 3. This gives us the point (0, 3). Similarly, set y = 0, and solve for x: -x + 2(0) = 6? x = -6. This gives us the point (-6, 0).
4. Plot the line 1x + 2y = 2 using the same method. Set x = 0: 1(0) + 2y = 2? y = 1. This gives us the point (0, 1). Set y = 0: 1x + 2(0) = 2? x = 2. This gives us the point (2, 0).
Now, we have plotted all four lines. To find the feasible region, shade in the region that satisfies all of the inequalities:
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The shaded region represents the feasible region.
To find the coordinates of each vertex, which are the points where two or more lines intersect, we need to solve the system of equations formed by the lines' equations.
Let's find the points of intersection:
1. Intersection of lines y = 1 and x = 2: The point (2, 1).
2. Intersection of lines x = 2 and -x + 2y = 6: Solving the equations, we get x = 2 and y = 4. So, the point (2, 4).
3. Intersection of lines -x + 2y = 6 and 1x + 2y = 2: Solving the equations, we get x = 4 and y = 1. So, the point (4, 1).
Now we have the coordinates of each vertex of the feasible region:
- Point 1: (2, 1)
- Point 2: (2, 4)
- Point 3: (4, 1)