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4. Graphs of Linear Inequalities Solving Systems of Linear Inequalities Application of Systems of Inequalities: Linear Programming Systems of Inequalities and Linear Programming Systems of Equations and Matrices > Systems of Inequalities and Linear Programming Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra?campaign_content=book_196_section_47&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=dir ect&utm_source=boundless

5. 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. Graphs of Linear Inequalities Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/systems-of-inequalities-and-linear- programming-47/graphs-of-linear-inequalities-209- 5516?campaign_content=book_196_section_47&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess View on Boundless.com Systems of Equations and Matrices > Systems of Inequalities and Linear Programming

6. 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. Solving Systems of Linear Inequalities Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/systems-of-inequalities-and-linear- programming-47/solving-systems-of-linear-inequalities-210- 8062?campaign_content=book_196_section_47&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess View on Boundless.com Systems of Equations and Matrices > Systems of Inequalities and Linear Programming

7. 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 Systems of Equations and Matrices > Systems of Inequalities and Linear Programming Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/systems-of-inequalities-and-linear- programming-47/application-of-systems-of-inequalities-linear-programming-211- 11084?campaign_content=book_196_section_47&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boun dless Equation 1 View on Boundless.com Equation 2 View on Boundless.com

8. 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. Application of Systems of Inequalities: Linear Programming Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/systems-of-inequalities-and-linear- programming-47/application-of-systems-of-inequalities-linear-programming-211- 11084?campaign_content=book_196_section_47&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boun dless View on Boundless.com Systems of Equations and Matrices > Systems of Inequalities and Linear Programming

9. Free to share, print, make copies and changes. Get yours at www.boundless.com Appendix

10. 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 ≥, 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. Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices

11. 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. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph:." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-1446ece6-e0b3-475b-965d-4d4154c4532f View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-1446ece6-e0b3-475b-965d-4d4154c4532fView on Boundless.com Systems of Equations and Matrices

12. 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. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: System of Linear Equations." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-system-of-linear-equations View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-system-of-linear-equationsView on Boundless.com Systems of Equations and Matrices

13. 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. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-48e0f2ec-4565-467a-a02e-40bbca762049 View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-48e0f2ec-4565-467a-a02e-40bbca762049View on Boundless.com Systems of Equations and Matrices

14. 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". Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: Linear Inequality System With No Solutions." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-linear-inequality- system-with-no-solutions View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-linear-inequality- system-with-no-solutionsView on Boundless.com Systems of Equations and Matrices

15. 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". Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph:." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-77acb8bb-d7be-4526-985e-93c856030dff View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-77acb8bb-d7be-4526-985e-93c856030dffView on Boundless.com Systems of Equations and Matrices

16. 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. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: Gas Mileage Example." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-gas-mileage-example View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-gas-mileage-exampleView on Boundless.com Systems of Equations and Matrices

17. Interactive Graph: Graph of Single Inequality Graph of single inequality Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: Graph of Single Inequality." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-graph-of-single-inequality View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-graph-of-single-inequalityView on Boundless.com Systems of Equations and Matrices

18. Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices If the inequality y<2x+3 was graphed, which of the following points would fall in the shaded region? A) (-1,4) B) (-4,1) C) (1,4) D) (1,7)

19. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Systems of Equations and Matrices If the inequality y<2x+3 was graphed, which of the following points would fall in the shaded region? A) (-1,4) B) (-4,1) C) (1,4) D) (1,7)

20. Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices If the system 2x-y>-3 and 4x+y<5 was graphed, which of the following points would fall in the shaded region? A) (0,4) B) (0,2) C) (4,0) D) (-2,0)

21. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Systems of Equations and Matrices If the system 2x-y>-3 and 4x+y<5 was graphed, which of the following points would fall in the shaded region? A) (0,4) B) (0,2) C) (4,0) D) (-2,0)

22. Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices For the following system of linear inequalities, which of the following points falls in the shaded region: 2x-3y 0? A) (6,-2) B) (2,6) C) (6,2) D) (2,-6)

23. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Systems of Equations and Matrices For the following system of linear inequalities, which of the following points falls in the shaded region: 2x-3y 0? A) (6,-2) B) (2,6) C) (6,2) D) (2,-6)

24. Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices Why would one would choose to solve a system of inequalities using a non-graphical method? A) All of these answers. B) Graph paper is not available. C) There are too many variables. D) There are too many inequalities.

25. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Systems of Equations and Matrices Why would one would choose to solve a system of inequalities using a non-graphical method? A) All of these answers. B) Graph paper is not available. C) There are too many variables. D) There are too many inequalities.

26. Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices Find the maximum value of z=3a+2b+4c subject to the following constraints: 3a+2b+5c≤18, 4a+2a+3c≤16, and 2a+b+c≥4. A) z=16 B) z=17 C) z=18 D) z=19

27. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Systems of Equations and Matrices Find the maximum value of z=3a+2b+4c subject to the following constraints: 3a+2b+5c≤18, 4a+2a+3c≤16, and 2a+b+c≥4. A) z=16 B) z=17 C) z=18 D) z=19

28. Attribution Wikibooks. "Algebra/Graphing Systems of Inequalities." CC BY-SA 3.0 http://en.wikibooks.org/wiki/Algebra/Graphing_Systems_of_InequalitiesCC BY-SA 3.0 http://en.wikibooks.org/wiki/Algebra/Graphing_Systems_of_Inequalities Wikibooks. "Applicable Mathematics/Linear Programming and Graphical Solutions." CC BY-SA 3.0 http://en.wikibooks.org/wiki/Applicable_Mathematics/Linear_Programming_and_Graphical_SolutionsCC 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_InequalitiesCC BY-SA 3.0http://en.wikibooks.org/wiki/Algebra/Graphing_Inequalities Wiktionary. "linear." CC BY-SA 3.0 http://en.wiktionary.org/wiki/linearCC BY-SA 3.0http://en.wiktionary.org/wiki/linear Wiktionary. "inequality." CC BY-SA 3.0 http://en.wiktionary.org/wiki/inequalityCC BY-SA 3.0http://en.wiktionary.org/wiki/inequality Wikipedia. "System of inequalities." CC BY-SA 3.0 http://en.wikipedia.org/wiki/System_of_inequalitiesCC BY-SA 3.0http://en.wikipedia.org/wiki/System_of_inequalities Wikipedia. "Linear inequality." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Linear_inequalityCC BY-SA 3.0http://en.wikipedia.org/wiki/Linear_inequality Wiktionary. "subset." CC BY-SA 3.0 http://en.wiktionary.org/wiki/subsetCC BY-SA 3.0http://en.wiktionary.org/wiki/subset Wiktionary. "mutually exclusive." CC BY-SA 3.0 http://en.wiktionary.org/wiki/mutually+exclusiveCC BY-SA 3.0http://en.wiktionary.org/wiki/mutually+exclusive Wikipedia. "Linear programming." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Linear_programmingCC BY-SA 3.0http://en.wikipedia.org/wiki/Linear_programming Wikibooks. "Applicable Mathematics/Linear Programming and Graphical Solutions." CC BY-SA 3.0 http://en.wikibooks.org/wiki/Applicable_Mathematics/Linear_Programming_and_Graphical_SolutionsCC BY-SA 3.0 http://en.wikibooks.org/wiki/Applicable_Mathematics/Linear_Programming_and_Graphical_Solutions Wikibooks. "A-level Mathematics/OCR/D1/Linear Programming." CC BY-SA 3.0 http://en.wikibooks.org/wiki/A- level_Mathematics/OCR/D1/Linear_ProgrammingCC BY-SA 3.0http://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+functionCC BY-SA 3.0http://en.wiktionary.org/wiki/objective+function Wiktionary. "constraint." CC BY-SA 3.0 http://en.wiktionary.org/wiki/constraintCC BY-SA 3.0http://en.wiktionary.org/wiki/constraint Wikipedia. "Simplex algorithm." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Simplex_algorithmCC BY-SA 3.0http://en.wikipedia.org/wiki/Simplex_algorithm Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices