SOLUTION: Linear Inequalities The school board is investigating ways to hire a faculty for the summer school program. They can hire teachers and aids. A minimum of 20 faculty members is ne

Question 41430: Linear Inequalities
The school board is investigating ways to hire a faculty for the summer school program. They can hire teachers and aids. A minimum of 20 faculty members is needed to run the program, and there must be at least 12 teachers. For a proper teacher-to-aide ratio, The number of aids must be no more than twice the number of teachers. There can be no more than 50 faculty members altogether. (Note there cannot be a negative number of teachers or aides)
a) Write an inequality for each condition given in this situation.
b) Graph the inequalities on the same coordinate axis.
c) If the school board hires 12 teachers and 5 aides, will all of the conditions be satisfied?
Thank You
Answer by psbhowmick(878) (Show Source): You can put this solution on YOUR website!
Let the number of teachers be 'x' and that of aides be 'y'.
As the number of faculty has to be at least 20, so x + y >= 20.
As the number of faculty can to be at most 50, so x + y <= 50.
As there must be at least 12 teachers, so x >= 12.
Also the number of aides cannot exceede twice the number of teachers, so y <= 2x.
Let us first plot the graph.

AB represents x + y = 50
CD represents x + y = 20
OE represents y = 2x
FG represents x = 12

As the number of faculty has to be at least 20, so x & y must lie on the side of AB that doesn't contain the origin.
As the number of faculty has to be at least 50, so x & y must lie on the side of AB that contains the origin.
Thus x and y can take any integral value within or on the trapezium ABDC.

[Note: x & y can take only positive integral values as they are no. of persons which can only be positive integers.]

[Note: As the signs are >= or <= and not simply > or < so the points on the lines AB and CD also satisfy the given conditions. Hence the term 'on' has been used. 'Within' refers to the region inside ABCD and 'on' refers to points on line segments AB, BD, CD and DA.]

As the number of aides cannot exceede twice the number of teachers, so x and y can take any integral value within or on the trapezium IECA.
As there must be at least 12 teachers, so x and y can take any integral value within or on the pentagon AJHEC.

Now, the point P (12,5) represents the condition when 12 teachers and 5 aides are hired.
As P is outside the pentagon AJHEC, so the condition when 12 teachers and 5 aides are hired does not satisfy all the conditions.
[Specifically, this does not satisfy the condition that the number of faculty hired must be greater than or equal to 20 as 12 + 5 = 17 < 20.]
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