Graphs of Linear Inequalities When the equal sign in a linear equation is replaced with an inequality sign, a linear inequality is formed. Solutions of linear inequalities are ordered pairs. Example Solution Determine whether (1, 5) and (6, –2) are solutions of the inequality 3x – y < 5. The pair (1, 5) is a solution of the inequality, but (6, –2) is not. 3(1) – 5 5 –2 < 5TRUE 3x – y < 5 3(6) – (–2) 5 20 < 5FALSE 3x – y < 5

Example Solution Determine whether (1, 5) and (6, –2) are solutions of the inequality 3x – y < 5. The pair (1, 5) is a solution of the inequality, but (6, –2) is not. 3(1) – 5 5 –2 < 5TRUE 3x – y < 5 3(6) – (–2) 5 20 < 5FALSE 3x – y < 5

The graph of a linear equation is a straight line. The graph of a linear inequality is a half-plane, with a boundary that is a straight line. To find the equation of the boundary line, we simply replace the inequality sign with an equals sign. Example Graph Solution First: Graph the boundary y = x. Since the inequality is greater than or equal to, the line is drawn solid and is part of the graph of x y y = x Second: We choose a test point on one side of the boundary, say (0, 1). Substituting into the inequality we get We finish drawing the solution set by shading the half-plane that includes (0, 1). (0, 1)

Examples Graph the following inequalities.

Systems of Linear Inequalities To graph a system of equations, we graph the individual equations and then find the intersection of the individual graphs. We do the same thing for a system of inequalities, that is, we graph each inequality and find the intersection of the individual graphs.

Example The graph of the system

Let’s look at 6 different types of problems that we have solved, along with illustrations of each type. Type Example Solution Linear equations 2x – 8 = 3(x + 5) A number in one variable Graph Type Example Solution Linear inequalities –3x + 5 > 2 A set of numbers; in one variable an interval Graph

Type Example Solution Linear equations 2x + y = 7 A set of ordered in two variables pairs; a line Graph

Type Example Solution Linear inequalities x + y ≥ 4 A set of ordered in two variables pairs; a half-plane Graph

Type Example Solution System of x + y = 3, An ordered pair or equations in 5x – y = –27 a (possibly empty) two variables set of ordered pairs Graph

Type Example Solution System of 6x – 2y ≤ 12, A set of ordered inequalities in y – 3 ≤ 0, pairs; a region two variables y ≥  x of a plane Graph