## Systems of Inequalities and Linear Programming Resources

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Systems of Inequalities and Linear Programming

• Graphs of Linear Inequalities

• Solving Systems of Linear Inequalities

• Application of Systems of Inequalities: Linear Programming

Graphs of Linear Inequalities

• To graph a single linear inequality, first graph the inequality as if it were an equation.If the sign is ≤ or ≥, graph a normal line.If it is > or <, then use a dotted or dashed line.Then, shade either above or below the line, depending on if y is greater or less than mx + b.

• If there are multiple linear inequalities, then where all the shaded areas of each inequality overlap is the solutions to the system.

• If the shaded areas of all inequalities in a system do not overlap, then the system has no solution.

Solving Systems of Linear Inequalities

• To solve a system graphically, draw and shade in each of the inequalities on the graph, and then look for an area in which all of the inequalities overlap, this area is the solution.

• If there is no area in which all of the inequalities overlap, then the system has no solution.

• To solve a system non-graphically, find the intersection points, and then find out relative to those points which values still hold for the inequality.Narrow down these values until mutually exclusive ranges (no solutions) are found, or not, in which the solution is within your final range.

Application of Systems of Inequalities: Linear Programming

•The standard form for a linear program is: minimize [Equation 1], subject to [Equation 2].c is the coefficients of the objective function, x is the variables, A is the left-side of the constraints and b is the right side.

Application of Systems of Inequalities: Linear Programming

• The Simplex Method involves choosing an entering variable from the nonbasic variables in the objective function, finding the corresponding leaving variable that maintains feasibility, and pivoting to get a new feasible solution, repeating until you find a solution.

• In the Simplex Method, if there are no positive coefficients corresponding to the nonbasic variables in the objective function, then you are at an optimal solution.

• In the Simplex Method, if there are no choices for the leaving variable, then the solution is unbounded.

Appendix

Key terms

• constraint A condition that a solution to a problem must satisfy.

• inequality A statement that of two quantities one is specifically less than or greater than another.Symbols: < or ≤ or > or ≥, as appropriate.

• linear Of or relating to a class of polynomial of the form y = ax + b .

• mutually exclusive Describing multiple events or states of being such that the occurrence of any one implies the non-occurrence of all the others.

• objective function A function to be maximized or minimized in optimization theory.

• subset With respect to another set, a set such that each of its elements is also an element of the other set.

Interactive Graph: Linear Inequalities

The graph of several linear inequalities, where each is less than or equal to some value.The region where all of them overlap is considered the “feasible region”.

Interactive Graph: Gas Mileage Example

Graph of the inequality equations (red) and $y<32x$ (blue).Here is an example showing in what range a car can drive given a city mpg and a highway mpg, as shown by the overlapping section.

Interactive Graph: Graph of Single Inequality

Graph of single inequality

Interactive Graph: Linear Inequality System With No Solutions

Graph of the three inequalities $y\ge 1$ (red), $y\ge 2x+2$ (blue), and $y\le x+1$ (purple).This is a graph showing a system of linear inequalities that has no solution as there is no point in which the areas of all three inequalities overlap.Contrast this with the graph “Solution to Linear Inequality System”.

Interactive Graph: Solution to Linear Inequality System

Graph of the three inequalities $y\ge -2x-1$ (red), $y\ge 2x+1$ (blue), and $y\le x+2$ (purple).This is a graph showing the solutions to a linear inequality system.Note that it is the overlapping areas of all three linear inequalities.

Interactive Graph: System of Linear Equations

Graph of system of linear equations (red) and (blue).In a system of linear equalities, the solution is at the point of intersection.In comparison, the solution to a system of linear inequalities is defined by the space according to the inequalities.

Interactive Graph: Graph of Linear Inequality

Graph of the three inequalities $y<-3x+5$ (red), $y<-x+3$ (blue), and $y<-2x+10$ (purple).The overlapping area is the feasible region for a linear program with the constraints forming the lines.The optimal solution will occur at one of the intersection points of the lines.

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• Wikibooks. “Applicable Mathematics/Linear Programming and Graphical Solutions.” CC BY-SA 3.0 http://en.wikibooks.org/wiki/Applicable_Mathematics/Linear_Programming_and_Graphical_Solutions

• Wikibooks. “Algebra/Graphing Inequalities.” CC BY-SA 3.0 http://en.wikibooks.org/wiki/Algebra/Graphing_Inequalities

• Wiktionary. “linear.” CC BY-SA 3.0 http://en.wiktionary.org/wiki/linear

• Wiktionary. “inequality.” CC BY-SA 3.0 http://en.wiktionary.org/wiki/inequality

• Wikipedia. “System of inequalities.” CC BY-SA 3.0 http://en.wikipedia.org/wiki/System_of_inequalities

• Wikipedia. “Linear inequality.” CC BY-SA 3.0 http://en.wikipedia.org/wiki/Linear_inequality

• Wiktionary. “subset.” CC BY-SA 3.0 http://en.wiktionary.org/wiki/subset

• Wiktionary. “mutually exclusive.” CC BY-SA 3.0 http://en.wiktionary.org/wiki/mutually+exclusive

• Wikipedia. “Linear programming.” CC BY-SA 3.0 http://en.wikipedia.org/wiki/Linear_programming

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• Wikibooks. “A-level Mathematics/OCR/D1/Linear Programming.” CC BY-SA 3.0 http://en.wikibooks.org/wiki/A-level_Mathematics/OCR/D1/Linear_Programming

• Wiktionary. “objective function.” CC BY-SA 3.0 http://en.wiktionary.org/wiki/objective+function

• Wiktionary. “constraint.” CC BY-SA 3.0 http://en.wiktionary.org/wiki/constraint

• Wikipedia. “Simplex algorithm.” CC BY-SA 3.0 http://en.wikipedia.org/wiki/Simplex_algorithm

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