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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.
Figure 1. Graph of boundary line for y < x – 3.
figure
x
y
3
0
0
–3
4
1
Now shade the lower half-plane as shown in Figure 2, since y < x – 3.
Figure 2. Graph of inequality y < x – 3.
figure
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 xy ≤ 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.
Figure 3. Graph of the boundary line for y ≥ 2x.
figure
x
y
0
0
1
2
2
4
Since y ≥ 2 x, you should shade the upper half-plane. If in doubt, or to check, plug in a pair of coordinates. Try the pair (1, 1).
equation
So you should shade the half-plane that does not contain (1, 1) as shown in Figure 4.
Figure 4. Graph of inequality y ≥ 2 x.
figure

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Posted by Muhammad Atif Saeed on 23:42. Filed under . You can follow any responses to this entry through the RSS 2.0. Feel free to leave a response

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I am doing ACMA from Institute of Cost and Management Accountants Pakistan (Islamabad). Computer and Accounting are my favorite subjects contact Information: +923347787272 atifsaeedicmap@gmail.com atifsaeed_icmap@hotmail.com

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