Graphing Linear Inequalities Basic Mathematics Review 837 Linear inequalities pla an important role in applied mathematics. The are used in an area of mathematics called linear programming which was developed during World War II. In order to graph linear inequalities in two variables, we first have to graph the linear equation called the boundar line which divides the coordinate plane into half-planes. /-plane /-plane When ou graph the boundar line ou must have a wa to indicate whether the points on the line are part of the solution of the inequalit. If the line is a dashed line, the points are not included in the solution; however, if the boundar line is solid, then the points on the line are part of the solution. Dashed boundar line: < or >. Solid boundar line: or. Eample: Graph 3 + 6 Answer: Step : Graph the boundar line. To graph the boundar line, first solve the linear inequalit for, to get the inequalit in slope-intercept form. 3 + 6 3 + 3 6 3-3 + 6 Decide if the line should be solid or dashed. If the inequalit is 'less than or equal to' or 'greater than or equal to', then the points on the line are part of the solution, and the line should be solid. If the inequalit is 'less than' or 'greater than', then the points on the line are not part of the solution and the line should be dashed. Our eample has a 'less than or equal to', so its graph will have a solid line. Unlawful duplication is a violation of copright laws. Objective Review 837
Review 837 Basic Mathematics Graph the linear equation formed b substituting an equal sign for the inequalit sign. Graph = -3 + 6 as a solid line. Step : Choose a test point not on the boundar line and test if it is a solution of the inequalit. We substitute a test point that is not on the boundar line, b substituting the -coordinate for and the -coordinate for in our inequalit. If the inequalit we get after we simplif is true, then the half-plane that contains our test point, should be shaded. If the statement is false, then the other half-plane should be shaded. Let's test the point (0, 0) b substituting 0 for and 0 for into 3 + 6. 3 + 6 3(0) + (0) 6 0 + 0 6 0 6 We know that 0 is less than or equal to 6, so our statement is true. Therefore, we shade the half-plane that contains our test point, (0, 0). Unlawful duplication is a violation of copright laws. Objective Review 837
Basic Mathematics Review 837 Eample: Graph > +. Step : Graph the boundar line. Since the inequalit is >, the boundar line is dashed. The points on the boundar line are NOT part of the solution. Graph = + as a dashed line. Unlawful duplication is a violation of copright laws. Objective Review 837 3
Review 837 Basic Mathematics Step : Choose a test point not on the boundar line and test if it is a solution of the inequalit. We will test (0, 0) in > + 0 > (0) + 0 > Since the statement is false, we shade the half-plane that does not include the test point. Eample: Graph 3. Step : Graph the boundar line. Graph = 3 as a solid line. The graph of = 3 is a solid vertical line through the point (3, 0). Unlawful duplication is a violation of copright laws. Objective Review 837
Basic Mathematics Review 837 Step : Choose a test point not on the boundar line and test if it is a solution of the inequalit. If we test the point (0, 0) in 3 : 0 3 Since the result is a true statement, we shade the side of the vertical line containing the test point. We can see that all the points in the shaded region have -coordinates that are less than or equal to 3. Eample: Graph >. Step : Graph the boundar line. Graph = as a dashed line. The graph of = is a horizontal line through the point 0,. ( ( Unlawful duplication is a violation of copright laws. Objective Review 837
Review 837 Basic Mathematics Step : Choose a test point not on the boundar line and test if it is a solution of the inequalit Test the point (0, 0) in > : 0 > Since the result is a false statement, we shade the side of the vertical line that does NOT contain the test point. We can see that all the points in the shaded region have -coordinates that are greater than. As a check we can also test a point in the shaded region. If we test the point (0, ), for eample, we see that we get a true statement. Since =, >. Graphing Linear Inequalities Review Step Graph the boundar line. Dashed: < or > Solid: or Step Test a point not on the line. (0,0) is a good point to use (if it is not on the line Step 3 Step Shade one side of the boundar line. Check our solution. as it allows ou to evaluate the inequalit quickl If the test point is a solution to the inequalit, shade the side where the test point lies; otherwise, shade the other side. Unlawful duplication is a violation of copright laws. Objective Review 837 6