981 CHAPTER 9 Test Prep Concepts Examples Use a solid boundary for a nonstrict inequality 1… or Ú2 or a dashed boundary for a strict inequality 16 or 72. 9.6 Systems of Inequalities and Linear Programming Graphing an Inequality in Two Variables Method 1 If the inequality is or can be solved for y, then the following hold true. • The graph of y 6ƒ1x2 consists of all the points that are below the graph of y = ƒ1x2. • The graph of y 7ƒ1x2 consists of all the points that are above the graph of y = ƒ1x2. Method 2 If the inequality is not or cannot be solved for y, then choose a test point not on the boundary. • If the test point satisfies the inequality, then the graph includes all points on the same side of the boundary as the test point. • If the test point does not satisfy the inequality, then the graph includes all points on the other side of the boundary. Solving Systems of Inequalities To solve a system of inequalities, graph all inequalities on the same axes, and find the intersection of their solution sets. Solving a Linear Programming Problem Step 1 Write all necessary constraints and the objective function. Step 2 Graph the region of feasible solutions. Step 3 Identify all vertices (corner points). Step 4 Find the value of the objective function at each vertex. Step 5 The solution is given by the vertex producing the optimum value of the objective function. Fundamental Theorem of Linear Programming If an optimal value for a linear programming problem exists, then it occurs at a vertex of the region of feasible solutions. Graph y Ú x2 - 2x + 3. x 0 y 1 (1, 2) 3 Graph the solution set of the system. 3x - 5y 7 -15 x2 + y2 … 25 x y 3 5 5 0 The region of feasible solutions for the system below is given in the figure. x y (0, 0) (12, 0) (8, 3) (0, 7) x + 2y … 14 3x + 4y … 36 x Ú 0 y Ú 0 Maximize the objective function 8x + 12y. Maximum Vertex Value of 8x +12y 10, 02 8102 + 12102 = 0 10, 72 8102 + 12172 = 84 18, 32 8182 + 12132 = 100 112, 02 81122 + 12102 = 96 The maximum of 100 occurs at 18, 32. 9.7 Properties of Matrices Addition and Subtraction of Matrices To add (subtract) matrices of the same dimension, add (subtract) corresponding elements. Find the sum or difference. c 2 0 3 4 -1 9d + c -8 5 12 3 1 -3d = c -6 5 15 7 0 6d c 5 -8 -1 8d - c -2 3 4 -6d = c 7 -11 -5 14d
RkJQdWJsaXNoZXIy NjM5ODQ=