Content:  This document describes how to solve a linear programming problem with the Inequality App, Inequal.

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© 2013 Frank Kizer  

General:  This program omits the part of the application that calculates the value of the objective function  and replaces that with the option to either calculate those values by hand
      by substituting in the objective function or by using list arithmetic.

Procedure:  Use of the application will be demonstrated by using the problem that follows.
Find the maximum of the objective function z=2x +5y, subject to the following constraints:
3x+2y < 6 (Eq 1a)
-x+2y <  4 (Eq 2a)
x≥0, y≥0
a) First put the two-variable equations in slope-intercept form.
y <  -3/2x+3 (Eq 1b)
y <  1/2x+2 (Eq 2b)
b) Enter the right side of those equations opposite Y1 and Y2 respectively and enter 0 opposite Y3.
c) Set the WINDOW at Xmin = -1, Ymin=-1, Ymax= the largest value of "b" plus a few units, say 5 in this problem.  Set Xmax at least one unit higher that the value largest value of the
y-intercept of the inequalities with a negative slope. Say, three in this case. You might want to press GRAPH and see if all of the corner points of the bounded region are on the screen.

Now we will enter the inequality signs.
a) Move the cursor to the sign (either equal or inequality) after Y1. If the inequality symbols do not appear at the bottom of the screen, you will need to start the Inequality App. Do that by pressing
APPS, move the cursor down to Inequal, or Inequalz for the international version, and press ENTER, ENTER. The Y= editor screen should be displayed.
b) Place the cursor on the equal sign opposite Y1 and press ALPHA, F3 (
ZOOM). The equal sign should have been replaced by the inequality < . (See the completed entry in Figure 1 below.
c) Do the same for Y2; then opposite Y3, press ALPHA, F5 (GRAPH). The symbol >   should have  replaced the equal sign before the 0.

Figure 1:  Completed Y= screen.

d) Now, move the cursor up to the "X" in the upper left corner and press ENTER. With the cursor on the equal sign opposite X1, press ALPHA, F5 (GRAPH) to enter  > ;  then enter a zero after that symbol.


Figure 2:  Completed X= screen. 

e) Press GRAPH to draw and shade the graphs.  The screen in Figure 3 should appear.


Figure 3:  Graph with feasible region undefined.

f) Press ALPHA, F1, 1 to define the feasible region.


Figure 4: Defined feasible region.  Window settings: Xmin = -1; Xmax=3;Ymin=-1; Ymax=5

If you only want to graph, you may stop here.
At this time we will find the x- and y-values of the corner points. We will record the values for the corner points, so that we have the option of either calculating the value of the objective
function by hand or by using the lists.  
At first, getting the cursor to move to the location you want may seem a bit random.  But if you remember a few things, you’ll find it quite easy:  The first item in the expression in the upper left
of the screen tells what line you’re on. For example,  Y1∩Y2 tells you you’re on the first line in the Y= screen.  (Some browsers may return an empty square for the intersection symbol.) You
can use the left and right arrows to move to different points on that line.  If you want to move to line Y2, press the down arrow.  The up and down arrows move up and down according to
the order that the graphs or axes are listed on the Y= and X= screens.  Y= comes before X=.
a) So, press ALPHA, F3 (
ZOOM). If  Y1∩Y2 appears in the upper left of the screen, the values x=.5,  y=2.25 will be displayed at the bottom of the screen. (See Figure 5).  Record these values for
     later use.
 b) Press the right arrow to go to the point x=2, y=0.  
 c)  Now, press the down arrow to go back to the intersection point with Y2∩Y1 on the screen.  The Y2 in the expression Y2∩Y1 tells you that you’re on line Y2. Press the left arrow to go to
     the point x=0, y=2. Now press the left arrow to go to the last point, x=0, y=2.


Figure 5:  Screen prepared to go to the corner on the x-axis.

Evaluating the objective function:
Method by Hand:
At this point you can choose to plug the corner-point values into your calculator as follows: 2*.5 + 5*2.25; press ENTER and you’ll get 12.25 for the intersection point.  Do the other points

Method Using Lists:
a) Enter the values for x in list L1 and the y-values for y in L2.
b) Now, place the cursor over the name for L3 and enter the 2, 2nd, L1, +,5, 2nd, L2.


Figure 6:  Shows entry for L3 before pressing ENTER to do the calculations.

c) Press ENTER and the objective values will be entered in list L3.

Released: 9/15/12
Last Revised:

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