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Graphing Solution Sets for Inequalities

First, take a very simple inequality, y 1. The solution set consists of all points whose y-coordinate is greater than or equal to 1. These points are contained in the shaded region in the graph below.

This kind of region is called a half-plane because it is one of two parts of the plane into which a boundary line divides it. In this case, the region consists of all those points that lie on and above the line y = 1.

Another example is y 1. The “greater than or equal to” is changed to “less than or equal to”. The solution set for this inequality is shown below.

It is also a half-plane. In this case, the solution set consists of all points in the half-plane including and below the line y = 1.

In both cases, the equation of the boundary line is found by replacing the inequality symbol with an equals sign.

Consider y x. The equation of the boundary line is y = x , which is found by replacing the inequality symbol with an equals sign. The solution set for the inequality is the half-plane including and above the line. In the same way, the graph of the solution set for y x is the half-plane including and below this same line.

The solution sets for both inequalities are shown below.

The following general key idea is always true.

Key Idea

For any linear inequality, if the inequality symbol is replaced with an equals sign, the result is a line that divides the plane into two half-planes. The solution set for the inequality is one of these half-planes.

 

Example 1

Graph y 2x - 3.

Solution

First, replace the inequality symbol in y 2x - 3 with an equals sign, in this case y = 2x - 3. Graph the line.

Now, since the inequality states that the y-coordinate is greater than or equal to the linear expression in x , the solution set for the inequality is the set of points above this line.

This is shown in the shaded region above.

If the inequality symbol were reversed and the inequality was y 2x - 3, the solution set would be the set of points below this line, as shown below.

Key Idea

• If a linear inequality sets y greater than or equal to the linear expression in x, then the solution set is the set of points above the boundary line.

• If a linear inequality sets y less than or equal to the linear expression in x, then the solution set is the set of points below the boundary line.

Using this key idea, the solution set for y 5x - 7 is the set of points above the line y = 5x - 7. The solution set for y x - 1 is the set of points below the line y = x - 1.

So far, only inequalities containing and have been graphed. Inequalities with the symbols < and > are just as easy to graph.

Key Idea

• When the inequality symbol is or , draw a solid line on the boundary of the half-plane to indicate that the boundary line is included.

• When the inequality symbol is < or >, draw a dashed line on the boundary of the half-plane to indicate that the boundary line is not included.

This key idea is illustrated by the graphs shown below.

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