Download 7.1 Graphing Linear Inequalities in Two Variables

7.1 Graphing Linear Inequalities in Two Variables Examples of linear inequalities in two variables include 3
2
6,
4
12,
2
5
10. A solution of a linear inequality is any ordered pair (x,y) of real numbers that satisfies the inequality. For example 3, 2 is a solution of 3
2
6, since 3 3
2 2
6 is a true statement. A linear inequality has an infinite number of solutions. The best way to show these solutions is to sketch the graph of the inequality, which consists of all points in the plane whose coordinates satisfy the inequality. Guidelines for Graphing Linear Inequalities in Two Variables
 Replace the inequality symbol with an “=”, and sketch the corresponding boundary line: < or > boundary line is dashed and the boundary line is solid.  Substitute a test point (any point not on the boundary line) into the original inequality. If the inequality is true, then shade side containing the test point; otherwise shade the opposite side. Graph each of the following linear inequalities. 1. 2
5
10 Solution: Graph the solid boundary line 2
5
10 2
5
10 0 2 5 0
Substitute the test point (0,0) into the inequality 2
5
10 and get 2 0
5 0
10 which is a false statement. So all solutions lie opposite of side containing the test point. 2. 4
3
12 Solution: Graph the dashed boundary line 4
4
3
12 3
12 0
4 3 0
Substitute the test point (0,0) into the inequality 4
3
12 and get 4 0
3 0
12 which is a true statement, so all solutions lie on the side containing the test point. 2 3. Solution: Graph the dashed boundary line 2 . 2 Substitute the test point (1,1) into the 0
2
0 inequality 4
2 and get 1 > ‐2 (true) The graph of a system of inequalities is made up of all those points which satisfy all the inequalities of the system at the same time. Graph the system: 4
4. 2
Solution: Graph each solid boundary line: 4 2 0 4 0
2 4 0
2 0
Substitute the test point (0,0) Into each inequality and get: 0 0 4; 0 4;
0 0 2; 0 2;
2
2
1 5. 0
Solution: Graph all the dashed boundary lines 2,
2,
1,