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Using Boundless Presentations The Appendix The appendix is for you to use to add depth and breadth to your lectures. You can simply drag and drop slides from the appendix into the main presentation to make for a richer lecture experience. Free to edit, share, and copy Feel free to edit, share, and make as many copies of the Boundless presentations as you like. We encourage you to take these presentations and make them your own. Free to share, print, make copies and changes. Get yours at Boundless Teaching Platform Boundless empowers educators to engage their students with affordable, customizable textbooks and intuitive teaching tools. The free Boundless Teaching Platform gives educators the ability to customize textbooks in more than 20 subjects that align to hundreds of popular titles. Get started by using high quality Boundless books, or make switching to our platform easier by building from Boundless content pre-organized to match the assigned textbook. This platform gives educators the tools they need to assign readings and assessments, monitor student activity, and lead their classes with pre-made teaching resources. Get started now at: If you have any questions or problems please

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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 ect&utm_source=boundless

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 programming-47/graphs-of-linear-inequalities ?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

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 programming-47/solving-systems-of-linear-inequalities ?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

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 programming-47/application-of-systems-of-inequalities-linear-programming ?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

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 programming-47/application-of-systems-of-inequalities-linear-programming ?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

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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 Systems of Equations and Matrices

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 Boundless. "Interactive Graph:." CC BY-SA View on Boundless.comCC BY-SA 3.0https:// on Boundless.com Systems of Equations and Matrices

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 Boundless. "Interactive Graph: System of Linear Equations." CC BY-SA View on Boundless.comCC BY-SA 3.0https:// on Boundless.com Systems of Equations and Matrices

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 Boundless. "Interactive Graph." CC BY-SA View on Boundless.comCC BY-SA 3.0https:// on Boundless.com Systems of Equations and Matrices

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 Boundless. "Interactive Graph: Linear Inequality System With No Solutions." CC BY-SA system-with-no-solutions View on Boundless.comCC BY-SA 3.0https:// system-with-no-solutionsView on Boundless.com Systems of Equations and Matrices

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 Boundless. "Interactive Graph:." CC BY-SA View on Boundless.comCC BY-SA 3.0https:// on Boundless.com Systems of Equations and Matrices

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 Boundless. "Interactive Graph: Gas Mileage Example." CC BY-SA View on Boundless.comCC BY-SA 3.0https:// on Boundless.com Systems of Equations and Matrices

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

Free to share, print, make copies and changes. Get yours at 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)

Free to share, print, make copies and changes. Get yours at Boundless - LO. "Boundless." CC BY-SA BY-SA 3.0http:// 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)

Free to share, print, make copies and changes. Get yours at 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)

Free to share, print, make copies and changes. Get yours at Boundless - LO. "Boundless." CC BY-SA BY-SA 3.0http:// 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)

Free to share, print, make copies and changes. Get yours at 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)

Free to share, print, make copies and changes. Get yours at Boundless - LO. "Boundless." CC BY-SA BY-SA 3.0http:// 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)

Free to share, print, make copies and changes. Get yours at 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.

Free to share, print, make copies and changes. Get yours at Boundless - LO. "Boundless." CC BY-SA BY-SA 3.0http:// 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.

Free to share, print, make copies and changes. Get yours at 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

Free to share, print, make copies and changes. Get yours at Boundless - LO. "Boundless." CC BY-SA BY-SA 3.0http:// 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

Attribution Wikibooks. "Algebra/Graphing Systems of Inequalities." CC BY-SA BY-SA Wikibooks. "Applicable Mathematics/Linear Programming and Graphical Solutions." CC BY-SA BY-SA Wikibooks. "Algebra/Graphing Inequalities." CC BY-SA BY-SA 3.0http://en.wikibooks.org/wiki/Algebra/Graphing_Inequalities Wiktionary. "linear." CC BY-SA BY-SA 3.0http://en.wiktionary.org/wiki/linear Wiktionary. "inequality." CC BY-SA BY-SA 3.0http://en.wiktionary.org/wiki/inequality Wikipedia. "System of inequalities." CC BY-SA BY-SA 3.0http://en.wikipedia.org/wiki/System_of_inequalities Wikipedia. "Linear inequality." CC BY-SA BY-SA 3.0http://en.wikipedia.org/wiki/Linear_inequality Wiktionary. "subset." CC BY-SA BY-SA 3.0http://en.wiktionary.org/wiki/subset Wiktionary. "mutually exclusive." CC BY-SA BY-SA 3.0http://en.wiktionary.org/wiki/mutually+exclusive Wikipedia. "Linear programming." CC BY-SA BY-SA 3.0http://en.wikipedia.org/wiki/Linear_programming Wikibooks. "Applicable Mathematics/Linear Programming and Graphical Solutions." CC BY-SA BY-SA Wikibooks. "A-level Mathematics/OCR/D1/Linear Programming." CC BY-SA 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 BY-SA 3.0http://en.wiktionary.org/wiki/objective+function Wiktionary. "constraint." CC BY-SA BY-SA 3.0http://en.wiktionary.org/wiki/constraint Wikipedia. "Simplex algorithm." CC BY-SA BY-SA 3.0http://en.wikipedia.org/wiki/Simplex_algorithm Free to share, print, make copies and changes. Get yours at Systems of Equations and Matrices