Chapter 9 Linear and Quadratic Inequalities

9.1 Linear Inequalities in Two Variables An inequality in the two variables x and y describes a region in the Cartesian plane. The set of points that satisfy a linear inequality can be called the solution region. A linear inequality can be in four forms: Ax + By < C Ax + By ≤ C Ax + By > C Ax + By ≥ C Where A, B, and C are real numbers. Graphing Inequalities: Graph the corresponding equation (use a solid line for inequalities that use ≥ or ≤. Otherwise use a dotted line.) Select a test point on one side of the line. Often (0,0) is the easiest choice. Insert this point into the inequality. Shade on the side of the line of the test point if it works, on the other side if it does not.

Example: 5 Graph and solve the inequality 4x - 5y < 20 4x - 5y = 20 - 5y = - 4x + 20 Divide both sides by negative 5. y = 4x - 4 5 Now select your test point. (0,0) is a good pick as long as it is not on the line. Plug in (0,0) Plug in (5,0) 4(0) - 5(0) < 20 4(5) - 5(0) < 20 0 < 20 20 < 20 True ✓ False x Now plot the point on the graph, and shade the solution region that contains this point.

9.2 Quadratic Inequalities in One Variable A quadratic inequality can be in four forms: ax2 + bx + c < 0 - you can solve graphically or algebraically ax2 + bx + c ≤ 0 - the solution in one variable can have no values, one ax2 + bx + c > 0 value, or an infinite number of values ax2 + bx + c ≥ 0 Example: Solve the inequality 6x2 + 11x - 10 < 0 6x2 + 11x - 10 = 0 - 5 0 2 (2x + 5)(3x - 2) = 0 2 3 x = - 5 x = 2 2 3 The solution of the inequality 6x2 + 11x - 10 < 0 is the values of x, for which the graph of f(x) = 6x2 + 11x - 10 lies below the x - axis. Therefore, the solution is - 5 < x < 2 2 3

9.3 Quadratic Inequalities in Two Variables A quadratic inequality can be in four forms: y < ax2 + bx + c y ≤ ax2 + bx + c y > ax2 + bx + c y ≥ ax2 + bx + c Example: Graph and solve the inequality y ≥ x2 - 1 Graph using a solid line. Plug in (0,0) to test the inequality Plug in (3,0) to test the inequality 0 ≥ 02 - 1 0 ≥ 32 - 1 0 ≥ - 1 0 ≥ 8 True ✓ False x Then shade the solution region that contains this point.