Solve systems of linear inequalities by graphing

Learning Objectives Graph a single linear inequality in two variables, correctly identifying the boundary line and shaded region. Differentiate between using a solid line (≤, ≥) and a dashed line (<, >) for the boundary. Apply the test point method to determine which half-plane to shade for an inequality. Graph a system of two or more linear inequalities on the same coordinate plane. Identify the feasible region (solution set) as the common area of intersection of all shaded half-planes. Determine the vertices of a bounded feasible region by solving systems of linear equations. Verify algebraically whether a given point is a solution to the system. How can a company determine the optimal number of products to manufacture to maximize profit while staying within budget...

TermDefinitionExample Linear InequalityA mathematical statement that relates two linear expressions using an inequality symbol (<, >, ≤, ≥). It describes a region on the coordinate plane.y ≤ 2x - 1 System of Linear InequalitiesA set of two or more linear inequalities in the same variables that are considered simultaneously.{ y > x - 3, y ≤ -x + 5 } Boundary LineThe line that separates the coordinate plane into two half-planes. It is the graph of the linear equation corresponding to the inequality.For the inequality y > 3x + 2, the boundary line is y = 3x + 2. Half-PlaneThe region of the coordinate plane on one side of a boundary line. The solution to a single linear inequality is always a half-plane.For y > 3x + 2, the solution is the half-plane above the line y = 3x + 2. F...

Boundary Line Rule For inequalities with < or >, use a dashed line (---). For inequalities with ≤ or ≥, use a solid line (—). A dashed line indicates that the points on the line are NOT part of the solution. A solid line indicates that the points on the line ARE part of the solution. Test Point Method for Shading 1. Choose a test point not on the boundary line (e.g., (0,0) if possible). 2. Substitute its coordinates into the original inequality. 3. If the resulting statement is true, shade the half-plane containing the test point. 4. If it is false, shade the other half-plane. This is the most reliable method to determine which side of the boundary line represents the solutions to the inequality. Solution Identification The solution to the system of inequalities...