In this lesson, our instructor Vincent Selhorst-Jones explains Systems of Linear Inequalities. The lesson begins with a review of inequality solutions, negative multiplication flips, and basic operations. Youll then go over linear equations with two variables, learn how to graph linear inequalities, and how to graph a system. Vincent also teaches about test points for shading and linear programming. The lesson concludes with four practice questions to help you master graphing.
Remember, we solve an inequality in much the same way we solve an equation. However, instead of giving just a single answer, an inequality has an infinite variety of solutions. Also, remember, when doing algebra with an inequality, multiplying or dividing
both sides by a negative number will flip the direction of the inequality sign.
When we graph the answers to a linear inequality, we shade in the entire region that satisfies the inequality. Furthermore, the boundary line changes depending on if the inequality is strict ( < or > ) or not strict ( ≤ or ≥
). If it is strict, the line is dashed, while if it is not strict, it is a solid line.
To help figure out where to shade, begin by just graphing the boundary of the inequality (as if it were an equation). Then, once you've drawn the boundary (with the appropriate solid or dashed line), test some point on one side of the boundary. If the
point satisfies the inequality, shade that side. If not, shade the other side.
When finding the solutions to a system of linear inequalities, it is best to graph each inequality in the system. When we worked with linear equalities (=), we could use substitution or elimination, but that was because we were solving for a single answer.
With a system of inequalities, though, we aren't going to get just one answer-we're going to have infinitely many (or none, if the shadings don't overlap). Thus, graph each inequality, shade each appropriately as above, and the solutions to the system
are where all the shadings overlap.
An excellent application of linear inequalities is linear programming. It helps us optimize systems and make the best choice given certain requirements. We start with a linear objective function that we are trying to maximize or minimize
(such as profit or cost). The objective function is based off some number of variables. The variables in the objective function also have various constraints (given as a system of linear inequalities). This gives a region of feasible solutions
that are allowed by the constraints. Our objective is to find the maximum (or minimum) of our objective function within those feasible solutions.
The theory of linear programming says that if there is a max/min for the objective function within the constraints, it occurs at a vertex ("corner") of the feasible solutions. Thus, to find the max/min of an objective function, we begin by finding
the locations of the vertices by solving for intersection points. Once we know all the vertices, we can try each of them in our objective function, and whichever is highest/lowest is our max/min.
Systems of Linear Inequalities
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.