3.1 Graphing Linear Inequalities Graphing linear inequalities in two variables: The solution set for an inequality in two variables is shown on the Cartesian coordinate system. Boundary lines divide the coordinate system into two-half planes. One half is the solution area; the other half is not. The solution side must be shaded to indicate which points satisfy the inequality. We ll start with something easy horizontal and vertical lines. Graph the inequality: y 3 on the left graph below. [Graph the horizontal line, y = 3, first. This should be a solid line, indicating that the points on this line satisfy the inequality (Greater than or equal to). Shade above the line, because values above 3 have y values greater than 3.] Now graph the inequality: x 1 on the right graph below. Shade to the left of the line, because points to the left of the vertical line have x values less than 1. Why are points ON this boundary line solutions?
Now let s try to graph direct variation where the two variables are proportional. Graph the inequality: y 3x on the left graph below. [Graph the line, y = 3x, first. This line has a y-intercept of zero and a slope of 3. Shade above the line, because values above 3 have y values greater than 3.] Now graph the inequality: line? x 2y on the right graph below. What is the slope of this What is the y-intercept? How can you decide which side to shade? Other linear inequalities are solved in a similar fashion. The boundary line is graphed with either a solid ( or ) line or a dashed (< or >) line, but our problems will always have or and will always be graphed with a solid line. Generally, two points are required to graph a line. When the equation is given in standard form, ax + by = c, the two easiest points to find are the x- and y- intercepts. Graph the inequality: 2x 4y 12. First find the intercepts of the boundary line, 2x 4y = 12. The x-intercept is the point where y = 0. The y-intercept is the point where x = 0. 2x 4(0) = 12 2x = 12 x = 6 (6, 0) 2(0) 4y = 12-4y = 12 y = -3 (0, -3)
Graph these two points (don t forget to label them!) below. Draw the boundary line with a (solid/dashed) line. Before we shade we must determine which side of the line gives the solution set. You can do this in one of two ways. The easiest way is to use a test point. Choose a point off the boundary line. Plug it into the inequality and if it is true, then that point and all the other points on that side of the inequality are the solution and should be shaded. If it makes a false statement, then the other side of the boundary line should be shaded. Commonly, we use the origin (0, 0) as the test point. Check the statement: 2(0) 4(0) 12 0 12 Zero is NOT greater than or equal to 12, so the origin is NOT a solution. Shade on the other side of the boundary line. An alternative method is to solve the inequality for y, and then evaluate the result. For example, solving the inequality for y gives y.5x 3. REMEMBER TO CHANGE THE DIRECTION WHEN YOU DIVIDE BY A NEGATIVE NUMBER! Therefore we should shade below the positively sloped line, because y values must be less than or equal to,, the y values on the line.
Now make up two inequalities in two variables, in standard form. Problem: Graph the inequality: Now solve your problems on the graph below. Don t forget to check whether the boundary line should be solid or dashed, and don t forget to shade.
Graphing the intersection of two linear inequalities: The intersection of two inequalities is the region that satisfies BOTH. If you treat each inequality separately, graphing boundary lines and shading, but on the SAME coordinate system (the same graph), the solution set is the area that was shaded both times. Graph the intersection of the inequalities: x 3 and y 1. Before you begin to graph this problem. Suggest one point that you think is a solution, that satisfies both inequalities. Label it A on your graph. Now graph the first inequality. Did you need a solid or a dashed line? Don t forget to shade. Now graph the second inequality on the same graph. Did you shade above or below the boundary line? Why? Find the intersection of the two inequalities (the area that you shaded both times). Shade this area one more time to CLEARLY show that you know that all the points in that section of the graph satisfy BOTH inequalities. Is your point A in this intersection area? [You should have shaded to the left of the vertical line and above the horizontal line.]
Now graph the solution set for the following problem. Graph the intersection of the inequalities: 2x+ 4y 8 and 3x 6y > 9. Make sure you label intercepts. Make up two problems. Graph the intersection on the graph below. Problem: Graph the intersection of the inequalities: