Systems of Inequalities in Two Variables Sec. 7.5a

Graph of an Inequality An ordered pair (a, b) of real numbers is a solution of an inequality in x and y if the substitution x = a and y = b satisfies the inequality. Example  Which of the given points is a solution to the given inequality?

Graph of an Inequality An ordered pair (a, b) of real numbers is a solution of an inequality in x and y if the substitution x = a and y = b satisfies the inequality. When we have found all the solutions, we have solved the inequality. The graph of an inequality in x and y consists of all pairs (x, y) that are solutions of the inequality.

Steps for Drawing the Graph of an Inequality in Two Variables 1. Draw the graph of the equation obtained by replacing the inequality sign by an equal sign. Use a dashed line if the inequality is. Use a solid line if the inequality is. 2. Check a point in each of the two regions of the plane determined by the graph of the equation. If the point satisfies the inequality, then shade the region containing the point.

Practice Problems Draw the graph of the given inequality. State the boundary of the region. (0, 3) 1. Graph the line y = 2x + 3 (–1.5, 0) (solid or dashed line?) 2. Choose a point not on the line  plug into the original inequality 3. Shade the appropriate region

Practice Problems Draw the graph of the given inequality. State the boundary of the region. (0, 3) The graph of a linear inequality in any of the following forms: (–1.5, 0) is a half-plane The graph of the line y = ax + b is the boundary of the region. Boundary line: y = 2x + 3 is included

Practice Problems Draw the graph of the given inequality. State the boundary of the region. (2, 0) Boundary line x = 2 is included (0, –3) Boundary line y = –3 is excluded

Practice Problems Draw the graph of the given inequality. State the boundary of the region. Boundary curve y = x – 3 is included 2 (0, –3)

Systems of Inequalities A solution of a system of inequalities in x and y is an ordered pair (x, y) that satisfies each inequality in the system. When we have found all the common solutions, we have solved the system of inequalities.  Graphically, we use the same techniques as when we were working with single inequalities!!!

Practice Problems Solve the given system. Corners: (–1.535, 2.357), (0.869, 0.754) Boundaries are excluded The solution is the set of all ordered pairs in the shaded region

Practice Problems Solve the given system. Corners: (0, 0), (4, 0), (0, 9/2), (10/3, 2) Boundaries are included The solution is the set of all ordered pairs in the shaded region

Linear Programming In management science, it is often required to maximize or minimize a linear function called an objective function. This is a linear programming problem. In two dimensions, the objective function takes the form f = ax + by, and is used with a system of inequalities, called constraints. The solution to a linear programming problem occurs at one of the vertex points, or corner points, along the boundary of the region.

Linear Programming Practice Problems Find the maximum and minimum values of the objective function f = 5x + 3y, subject to the constraints given by the system of inequalities. Start with a graph! (0,8) (0,3) (4,0) (9,0)

Linear Programming Practice Problems Find the maximum and minimum values of the objective function f = 5x + 3y, subject to the constraints given by the system of inequalities. Start with a graph! Next, find the corner points! Then evaluate f at the corner points! (9,0), (0,8), (3,2) (x, y) f (9, 0) 45 (0, 8) 24 (3, 2) 21 at (3, 2)none! (unbounded region!)

Linear Programming Practice Problems Find the maximum and minimum values of the objective function f = 5x + 8y, subject to the constraints given by the system of inequalities. Where’s the graph? (0,10) (0,14/3) (5,0) (7,0)

Linear Programming Practice Problems Find the maximum and minimum values of the objective function f = 5x + 8y, subject to the constraints given by the system of inequalities. Where’s the graph? Corner points? Evaluation of f ? (0,0), (0,14/3), (5,0), (4,2) (x, y) f (0, 0) 0 (0, 14/3) 112/3 (5, 0) 25 (4, 2) 36 at (0, 0) at (0, 14/3)

Linear Programming Practice Problems Find the maximum and minimum values of the objective function f = 3x – 2y, subject to the constraints given by the system of inequalities. Where’s the graph? (0,8) (0,1) (0,10) (2,0)(3,0) (10,0)

Linear Programming Practice Problems Find the maximum and minimum values of the objective function f = 3x – 2y, subject to the constraints given by the system of inequalities. Where’s the graph? Corner points? Evaluation of f ? (2/5,8), (7,8), (90/49,40/49), (10/3,2/3) (x, y) f (2/5, 8) –14.8 (7, 8) 5 (90/49, 40/49)(10/3, 2/3) at (2/5,8) at (10/3,2/3) 190/4926/3