



Sometimes
you are handed a bunch of inequalities 


to
graph all at once. 





A
group of inequalities like this 


are
usually called constraints. 





Maybe
you have something like ... 





X + Y
£
5 


X
³
1 


Y
³
0 


X
£
4 





Here's
what you do ... 





1)
Choose any one of the inequalities and graph it. 


Shade
the correct side of the line VERY LIGHTLY 


Check
the side to shade using the point (0,0) if possible. 











2)
Choose a different inequality and graph it 


on
the same set of axis. 


Shade
the correct side VERY LIGHTLY. 


Check
using the point (0,0) if possible ... 











3)
Erase the shading in any area shaded 


by
only one of the two inequalities. 


What
is left is the area shaded by both inequalities. 


Also
find the point where the two lines cross.












4)
Choose another inequality and graph it 


on
the same set of axis. Shade the correct side. 











5)
Leave shaded only that area shaded 


by
all of the inequalities. 


Find
any new points where 2 lines cross. 











6)
Repeat the process for all other inequalities in the group. 


Here,
there is only one more left to do ... 











When
you do one of these, if everything works out right, 


you
get some shaded shape that has borders on all sides, 


and
labels on all of the corners. 





Well
that's just wonderful! 





BUT
WHAT IS IT GOOD FOR??? 





OK,
You might see a word problem 


that
goes something like this ... 





Example: 





Maximize
the value of 3X + 2Y subject to these constraints ... 





X + Y
£
5 


X
³
1 


Y
³
0 


X
£
4 





What
we need to do is find the maximum value 


that
we can get from 3X + 2Y. 





The
problem is that the values we choose for X and Y 


must
make all of the constraints true. 


That
means they are somewhere 


in
the shaded area of the graph. 





That's
what we were doing so far on this page. 


Finding
all of the values of X and Y 


that
make all 4 constraints true. 





The
problem is that even a small area 


contains
a heck of a lot of points. 





If
we didn't have any more help 


the
problem could still take forever. 





We
do get more help. 


Here
it is ... 





THE
ANSWER TO ONE OF THESE TYPE OF PROBLEMS 


IS
ALWAYS ONE OF THE CORNER POINTS! 





They
talk about why in later math courses, 


for
now, just be glad it's true and use it! 





We've
already figured out what the corner points are. 


Now
we just plug them into 3X + 2Y and see which one 


gives
us the biggest answer ... 





3X
+ 2Y 

(1,
0) 
3(1)
+ 2(0) = 3 
(1,
4) 
3(1)
+ 2(4) = 11 
(4,
0) 
3(4)
+ 2(0) = 12 
(4,
1) 
3(4)
+ 2(1) = 14 






So
the greatest value (14) happens when X = 4 and Y = 1. 





SOME
VARIATIONS 





1)
Instead of finding the greatest value, 


we
might want to find the least value. 





The
answer to this one is still one of the corner points 


of
the constraints. 


Just
plug them all in and see which one 


gives
you the smallest answer. 





For
example, the point X = 1, Y = 0 


gives
the least value for 3X + 2Y 


when
subject to our constraints. 





2)
The inequality used by some constraints 


might
be >
or <. 





When
this happens, do all of the same steps 


that
we did in the example. 


Use
dotted lines instead of solid lines 


for
any inequality using >
or <. 


Find
all of the corner points and check them. 


If
the answer is a corner point formed 


using
one or two dotted lines, 


THE
PROBLEM HAS NO ANSWER. 





3)
If the constraints don't form an enclosed shape, 


but
an open one like this ... 











A)
Find the value of the corner points we do have. 





B)
Pick some point on each of the open lines. 











If
a_{1} or b_{1} is a better answer than any of the
corner points 


THE
PROBLEM HAS NO ANSWER. 


If
neither a_{1} or b_{1} is a better answer, 


go
to the next step. 





C)
Pick two more points further down the open lines ... 











If
a_{1} is a better answer than a_{2} 


and
b_{1} is a better answer than b_{2} 


the
answer is the best corner point. 





If
a_{2} is a better answer than a_{1} 


OR
b_{2} is a better answer than b_{1} 


THE
PROBLEM HAS NO ANSWER. 





copyright 2005 Bruce Kirkpatrick 
