© Jill Zarestky
Math 141 Week in Review
Week in Review 4 Key Topics
3.1 Graphing Systems of Linear Inequalities in Two Variables
To graph an inequality or system of inequalities:
• Replace the inequality signs with equal signs and graph each line. These are the boundary
lines.
• The inclusion of each line in the solution region depends on the type of inequality:
o If the inequality uses ≥ or ≤, draw the boundary line as solid. The line is included.
o If the inequality uses > or <, draw the boundary line as dashed. The line is not
included.
• Each line divides the plane into two halves. One side of the line is the solution region of
the inequality.
• Pick a test point in one of the regions and plug it into the inequality to see if it makes the
inequality true. If it makes the inequality TRUE, then this region IS part of the solution. If it
makes the inequality FALSE, then this region is NOT part of the solution.
• Shade as appropriate. Note: Some instructors want you to shade the solution region and
some want you to shade the false region. Be sure you know what is expected of you and
always label your solution region as such.
• The solution set to a system of inequalities is the region that makes every inequality TRUE,
called the feasible region, feasible set, solution region, or solution set.
• The solution set of a system of linear inequalities is bounded if you can draw a circle around
it. Otherwise, it is unbounded.
3.2-3 Linear Programming
A linear programming problem has two parts but you should always define your variables first.
• A linear objective function to be maximized or minimized
• Constraints to which the objective function is subject
o Always a system of linear inequalities
o Look for key phrases such as “no more than” and “at least.”
Solve by using the Method of Corners
1. Graph the feasible set S.
2. Find the exact coordinates of all corner points (vertices) of S.
3. Evaluate the objective function at each corner point.
4. Find the vertex that renders the objective function a maximum (or minimum).
• If there is only one such vertex, then this vertex is a unique solution to the problem.
• If the objective function is maximized (or minimized) at two adjacent corner points of S,
there are infinitely many optimal solutions given by the points on the line segment
determined by these two vertices.
Possible Outcomes
• The feasible region is bounded so there is both a minimum and a maximum.
• The feasible region is unbounded. There will usually be a minimum but no maximum.
Check your objective function to be sure.
• The feasible region is the empty set. There is no solution.
Leftover resources are the differences between how much you actually use and how much was
available. Determine this by plugging your solution corner point back into the constraints.