MATH 1324 (Finite Mathematics or Business Math I) Lecture Notes
Author / Copyright: Kevin Pinegar
MATH 1324 Module 3 Notes: SYSTEMS OF INEQUALITIES & LINEAR PROGRAMMING
3.1 GRAPHING LINEAR INEQUALITIES
Steps To Graphing A Linear Inequality Of Two Variables
1. Graph the line Ax+By=C either as a solid line or dashed line. I call this line the boundary. If ≥
or ≤ use solid line; if < or > use a dashed line. (Solid line tells the viewer that all points on the
line are part of the solution. Dashed line tells viewer that the points on the line are not part of the
solution.)
2. Choose any point on the plane (grid) that is NOT on the boundary (line). Evaluate the inequality
at this point. (You will either get a true or false statement.)
3. If you get a true statement like (3≥0), then you shade the same side of the boundary that your
test point is on. If you get a false statement like (3<0), then you shade the side opposite of your
test point.
Example 1: a) Graph 2x-3y≤6
Inequality
and b) 2x-3y>6
Boundary
2x-3y=6
a) 2 x  3 y  6
2x-3y=6
b) 2 x  3 y  6
a) 2 x  3 y  6 Solid Line
{Shade the side (0,0) is on.}
Example 2: a) Graph 2x+3y>12
Inequality
Boundary
2x+3y=12
Intercepts
(0,-2), (3,0)
Test (0,0) & Conclusion
0≤6, True
(0,-2), (3,0)
0>6, False
b) 2 x  3 y  6 Dashed Line
{Shade opposite side of (0,0)}
and b) 2x+3y≤12
a) 2 x  3 y  12
2x+3y=12
b) 2 x  3 y  12
a) 2 x  3 y  6 Dashed Line
{Shade opposite side (0,0) is on.}
Intercepts
(0,4), (6,0)
Test (0,0) & Conclusion
0>12, False
(0,4), (6,0)
0≤12, True
b) 2 x  3 y  6 Solid Line
{Shade same side as (0,0)}
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MATH 1324 (Finite Mathematics or Business Math I) Lecture Notes
Author / Copyright: Kevin Pinegar
Vertical and Horizontal Boundaries
Example: a) x>3
b) y≤-1
a) x>3
b) y≤-1
Regions Restricted To Quadrants
Example: Graph a) x  0
a) x  0
b) y  0
c) x  0, y  0
b) y  0 (dashed line)
c) x  0, y  0
The last one was actually a primitive graph of a system of inequalities. In many applications
x  0, y  0 , because x and y represent items that cannot be negative. We will now move on to
systems of linear inequalities.
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MATH 1324 (Finite Mathematics or Business Math I) Lecture Notes
Author / Copyright: Kevin Pinegar
The boundary to this inequality is the equation (or line) 2x+3y=12. The inequalities 2x+3y>12 and
2x+3y<12 represent opposite sides of that line {called the upper and lower half-planes). You must
determine which region (upper or lower) the given inequality represents. This can be done by
selecting a test point (that is not on the line) and plugging it into the inequality. If the test point yields a
true statement, you shade the region that the test point is on. If the test point yields a false statement,
shade the side opposite the test point. Use (0,0) as your test point when possible.
1) Graph the boundary, 2x+3y=12, by finding the intercepts {(0,4) and (6,0)}
2) Plug the test point (0,0) into the inequality and determine if it yields a true or false statement.
2(0)+3(0)>12,
0>12, False
3) Determine which side to shade. {In this case, opposite the test point}
Solution: Since the test point is in the lower region, you should shade the region above the line. The
points on the line are not part of the solution, thus the line is drawn as a dashed line.
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MATH 1324 (Finite Mathematics or Business Math I) Lecture Notes
Author / Copyright: Kevin Pinegar
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