Question 1205026
<font color=black size=3>
x = number of full-time employees
y = number of part-time volunteers
x and y are integers
They must be nonnegative so {{{x >= 0}}} and {{{y >= 0}}}


Given facts<ol><li>The community center needs at least 5 full-time employees.</li><li>[They need] up to 9 part-time volunteers.</li><li>They want no more than 12 people. </li></ol>Fact 1 leads to {{{x >= 5}}}
Fact 2 leads to {{{y <= 9}}}
Fact 3 means {{{x+y <= 12}}}


The system of inequalities would be
{{{system(x >= 0,y >= 0,x >= 5,y <= 9,x+y <= 12)}}}


Because {{{x >= 0}}} and {{{y >= 0}}}, we focus on the upper right quadrant only.


Furthermore, we shade to the right of x = 5 because of {{{x >= 5}}}
We shade below y = 9 because of {{{y <= 9}}}
Lastly, shade under {{{x+y =12}}} to take care of {{{x+y <= 12}}}


The line {{{x+y =12}}} goes through (0,12) and (12,0)


All of those regions overlap to form a triangle with these corner points
A = (5,0)
B = (5,7)
C = (12,0)
Any point inside this region is a solution to the system. Points on the boundary are also solutions as well. Each boundary line is solid due to the "or equal to".


Desmos graph
<a href="https://www.desmos.com/calculator/bjoscdwkhm">https://www.desmos.com/calculator/bjoscdwkhm</a>
An example solution in the shaded region is (8,2) marked in blue.
A non-solution example is (8,6) marked in red.


The entire list of solutions is:
<pre>
(5,7), 
(5,6), (6,6)
(5,5), (6,5), (7,5)
(5,4), (6,4), (7,4), (8,4)
(5,3), (6,3), (7,3), (8,3), (9,3)
(5,2), (6,2), (7,2), (8,2), (9,2), (10,2)
(5,1), (6,1), (7,1), (8,1), (9,1), (10,1), (11,1)
(5,0), (6,0), (7,0), (8,0), (9,0), (10,0), (11,0), (12,0)
</pre>
There are 36 solution points listed above.
1 in the first row, 2 in the second, 3 in the third, and so on to get 1+2+3+4+5+6+7+8 = 36


If x or y were allowed to be non-integer values, then we'd have infinitely many solutions. 
</font>