Applied Mathematics for Business and Economics

Lecture Note Linear Programming (LP)

Ax By C+ <+≤+>+Ax By C Ax By C Ax By C≥

where A, B, and C are constants with at least one coefficients A, B not zero. If the
equality invelves < or >, the half-plane is open, and we use a dashed line to indicate
that the boundary line is not part of the solution set. If the inequalities involves
, the half-plane is closed and we use the solid line to indicate that the boundary

is part of the solution set.

≤≥ or

xy+< 24

Example 2
Graph the inequality

xy+=^24

xy+< 24
dashed line

Procedure for Graphing a Linear Inequality in Two Variables
To graph an inequality of one of the forms
Ax By C+ <+≤+>+Ax By C Ax By C Ax By C≥

we proceed as follows:

1. Graph the coresponding equationAx By C+ =. Use a dashed line
   for this line in the < and > cases, and a solid line in the ≤≥ and
   cases.


2. The solution set is the half-plane on one side of the boundary line.
   To determine which side, choose a test point Pab( , )not one the
   line and check to see if the coordinates a and b satisfy the given
   inequality.

Example 3
Graph each of the following inequalities:
(a) 2 xy−≥ 5 (b) 30 xy+ > (c) x< 4

1.2 Solving Systems of Linear Inequalities
To solve a system of linear inequalities such as

35

27

xy

xy

+ < 2

− ≥

or

4

32

23

xy

xy

xy

+ ≤

− <

+ >

we must find the set of all points (x,y)that satisfy all the inequalities in the system

simultaneously. In general this solution set will be a region of the plane, which we
shall refer to as the feasibility region for the system of inequalities. To obtain the
feasibility region of a given system of linear inequalities, we first gragph the
individual inequalities in the system on the same set of coordinate axes and then take
the intersection of these individual graphs.

Example 4