MATH 140 -- Intermediate Algebra -- Exercise Solutions
Section: 3.6

26. Graph x - 4y < 8
Answer: The equation of the boundary line is x - 4y = 8. Its graph is
                                              
Use (0, 0) as a test point. When the coordinates of (0, 0) are substitued into the inequality we obtain
0 - 4(0) < 8 a TRUE statement.
Therefore the solution set for x - 4y < 8 is the half-plane containing (0, 0). Because this is a strict inequality the boundary line is not part of the solution and is drawn as a dashed line.
The graph of x - 4y < 8 is:
                                              

28. Graph y ≥ - 2
Answer: The equation of the boundary line is y = - 2. Its graph is
                                             
Use (0, 0) as a test point. When the coordinates of (0, 0) are substitued into the inequality we obtain
0 ≥ -2 a TRUE statement.
Therefore the solution set for y ≥ - 2 is the half-plane containing (0, 0). Because this inequality permits equality the boundary line is part of the solution and is drawn as a solid line.
The graph of y ≥ - 2 is:
                                            

30. Graph - 3x + y ≤ 9
Answer: The equation of the boundary line is - 3x + y = 9. Its graph is
                                            
Use (0, 0) as a test point. When the coordinates of (0, 0) are substitued into the inequality we obtain
-3(0) + 0 ≤ 9 a TRUE statement.
Therefore the solution set for -3x + y ≤ 9 is the half-plane containing (0, 0). Because this inequality permits equality the boundary line is part of the solution and is drawn as a solid line.
The graph of - 3x + y ≤ 9 is:
                                          

32. Graph x + 2y > 0
Answer: The equation of the boundary line is x + 2y = 0. Its graph is
                                            
Use (1, 1) as a test point. When the coordinates of (1, 1) are substitued into the inequality we obtain
1 + 2(1) > 0 a TRUE statement.
Therefore the solution set for x + 2y > 0 is the half-plane containing (1, 1). Because this is a strict inequality the boundary line is not part of the solution and is drawn as a dashed line.
The graph of x + 2y > 0 is:
                                            

34. Graph 2x - 3y ≤ 9
Answer: The equation of the boundary line is 2x - 3y = 9. Its graph is
                                            
Use (0, 0) as a test point. When the coordinates of (0, 0) are substitued into the inequality we obtain
2(0) - 3(0) ≤ 9 a TRUE statement.
Therefore the solution set for 2x - 3y ≤ 9 is the half-plane containing (0, 0). Because this inequality permits equality the boundary line is part of the solution and is drawn as a solid line.
The graph of 2x - 3y ≤ 9 is:
                                             

36. Graph the union of x - y < 3 and x > 4
Answer: The equation of the boundary line of x - y < 3 is x - y = 3 and the boundary line of x > 4 is x = 4 Their graphs are
                                                  
Use (0, 0) as a test point. for both inequalities.
When the coordinates of (0, 0) are substituted into the blue inequality a true statement is obtained. Therefore the solution set for the blue inequality is the half-plane containing (0, 0). Because this is a strict inequality the boundary line is not part of the solution and is drawn as a dashed line.
When the coordinates of (0, 0) are substituted into the red inequality a false statement is obtained. Therefore the solution set for the red inequality is the half-plane which does not contain (0, 0). Because this is a strict inequality the boundary line is not part of the solution and is drawn as a dashed line. The individual graphs of the two inequalities are
                                                   

The union is both sets--the points shaded in blue as well as the points shaded in red. This is properly stated as the blue points OR the red points. Note the inclusive nature of how the word OR is used in mathematics.The graph of the union is obtained by combining the two graphs. The graph of the union is
                                           

38. Graph the intersection of y≥x and 2x - 4y ≥ 6
Answer: The equation of the boundary line of y≥x is x = y and the boundary line of 2x - 4y ≥ 6 is 2x - 4y = 6 Their graphs are

                                              
Use (2, 1) as a test point for the blue inequality to obtain 1 ≥ 2 a FALSE statement. Therefore the half-plane which does not contain the point (2, 1) is the solution to the blue inequality. Because this inequality permits equality the boundary line is part of the solution and is drawn as a solid line.
Use (0,0) as a test point for the red inequality to obtain 2(0) - 4(0) ≥ 6 which is a FALSE statement. Therefore the half-plane which does not contain the point (0, 0) is the solution to the red inequality. Because this inequality permits equality the boundary line is part of the solution and is drawn as a solid line. The individual graphs of the two inequalities are
                 
The intersection is the set of point common to both sets--the points shaded in blue AND shaded in red. The graph of the intersection is obtained by combining the two graphs. The graph of the intersection is the region in the following graph which is shaded bot red and blue.