y = the number of minibuses
the constraints are:
x >= 5
y >= 10
x + y <= 30
3x + y <= 54
x + y represents the total number vehicles.
3x + y represents the total number of units of garage space.
using the desmos.com calculator, you would graph the opposite of the inequalities.
specifically, you would graph.
x <= 5
y <= 10
x + y >= 30
3x + y >= 54
the area on the graph that is not shaded is the region of feasibility.
the constraints that satisfy the requirements of the problem will lie in the unshaded area of the graph represented by the inequality portion of the functions and on the lines of the graph represented by the equality portion of the functions.
the equality portion of the functions would be line on the graph of the following equations in this problem.
x = 5
y = 10
x + y = 30
3x + y = 54
the points are in (x,y) format.
x represents the number of buses.
y represents the number of minibuses.
you would look for whole values of x and y because the number of buses and minibuses have to be whole numbers.
i drew 3 graphs.
the first graph shows the corner points of the feasible region.
the second graph shows points that are in the feasible region.
the third graph shows points that are not in the feasible region.
here they are:
points that are on the corner points of the feasible region.
points that are in the feasible region and that are whole numbers.
points that are not in the feasible region and that are whole numbers.
all the points in the feasible region will satisfy all the constraints.
all the points not in the feasible region will not satisfy all the constraints.
please note that all of the constraints need to be satisfied for any given point to be in the feasible region.
if any of the constraints are not satisfied at a given point, then that point is not in the feasible region.
you will also notice that one of the corner points of the feasible region contains an x or a y that is not an integer.
that point is not valid because x and y both have to be whole numbers, since the number of buses and minibuses can't be a number that is not a whole number.
any point in the feasible region must satisfy all the constraints.
for example, the point (14,10) satisfies all the constraints.
x >= 5 becomes 14 >= 5 which is true.
y >= 10 becomes 10 >= 10 which is true.
x + y <= 30 becomes 14 + 10 <= 30 which become 24 <= 30 which is true.
3x + y <= 54 becomes 3 * 14 + 10 <= 54 which becomes 52 <= 54 which is true.
if a point does not satisfy ALL the constraints, then it is not in the feasible region.
for example, the point (15,10) does not satisfy all the constraints, even though it satisfies some of the constraints.
x >= 5 becomes 15 >= 5 which is true.
y >= 10 becomes 10 >= 10 which is true.
x + y <= 30 becomes 15 + 10 <= 30 which becomes 25 <= 30 which is true.
3x + y <= 54 becomes 3 * 15 + 10 <= 54 which becomes 55 <= 54 which is not true.
the point (15,10) satisfies some of the constraints, but does not satisfy all of the constraints.
therefore, it is not in the feasible region.
this can be seen on the graph because it is in the shaded area of the graph.
the shaded area of the graph is not in the feasible region.
the desmos.com calculator can be found at:
https://www.desmos.com/calculator