Published On:Thursday, 8 December 2011
Posted by Muhammad Atif Saeed
Linear Inequalities and Half-Planes
Linear Inequalities and Half-Planes
Each line plotted on a coordinate graph divides the graph (or plane) into two half-planes. This line is called the boundary line (or bounding line). The graph of a linear inequality is always a half-plane. Before graphing a linear inequality, you must first find or use the equation of the line to make a boundary line.Open half-plane
If the inequality is a “>” or “<”, then the graph will be an open half-plane. An open half-plane does not include the boundary line, so the boundary line is written as a dashed line on the graph.Example 1
Graph the inequality y < x – 3.First graph the line y = x – 3 to find the boundary line (use a dashed line, since the inequality is “<”) as shown in Figure 1.
x | y |
---|---|
3 | 0 |
0 | –3 |
4 | 1 |
To check to see whether you've shaded the correct half-plane, plug in a pair of coordinates—the pair of (0, 0) is often a good choice. If the coordinates you selected make the inequality a true statement when plugged in, then you should shade the half-plane containing those coordinates. If the coordinates you selected do not make the inequality a true statement, then shade the half-plane not containing those coordinates.
Since the point (0, 0) does not make this inequality a true statement,
y < x – 3 0 < 0 – 3 is not true.
You should shade the side that does not contain the point (0, 0). This checking method is often simply used as the method to decide which half-plane to shade.
Closed half-plane
If the inequality is a “≤”or “≥”, then the graph will be a closed half-plane. A closed half-plane includes the boundary line and is graphed using a solid line and shading.Example 2
Graph the inequality 2 x – y ≤ 0.First transform the inequality so that y is the left member.
Subtracting 2 x from each side gives
– y ≤ –2 x
Now dividing each side by –1 (and changing the direction of the inequality) gives y ≥ 2 x
Graph y = 2 x to find the boundary (use a solid line, because the inequality is “≥”) as shown in Figure 3. x | y |
---|---|
0 | 0 |
1 | 2 |
2 | 4 |
So you should shade the half-plane that does not contain (1, 1) as shown in Figure 4.