3.3 Graph Systems of Linear Inequalities

System of linear inequalities

A system of two or more linear inequalities in two variables

Solution of a system of linear inequalities

An ordered pair that is a solution of each inequality in the system

Graph of a system of linear inequalities

The graph of all solutions of the system

GRAPHING A SYSTEM OF LINEAR INEQUALITIES

To graph a system of linear inequalities, follow these steps:

Step 1 Graph each inequality in the system. You may want to use colored pencils to distinguish the different _half-planes_ .

Step 2 Identify the region that is _common_ to all the graphs of the inequalities. This region is the graph of the system. If you used colored pencils, the graph of the system is the region that has been shaded with _every_ color.

Example 1

Graph a system of two inequalities

Graph the system.

y < 2x + 1 Inequality 1

The graph of the system is the intersection of the shaded regions.

y £ -x - 3 Inequality 2

Graph each inequality in the system. Use different shades for each half-plane. Identify the region that is __common__ to both graphs.

Example 2

Graph a system with no solution

Graph the system.	y £ x+ 1	Inequality 1
	3x + 4y > 16	Inequality 2

The shaded regions __do not__ intersect.

Graph each inequality in the system. Use different shades for each half-plane. Identify the region that is common to both graphs. There is no region shaded __both colors__ . So, the system has __no solution__ .

Example 3

Graph a system with an absolute value inequality

Graph the system.	y £ 4	Inequality 1
	y > | x - 2 | + 1	Inequality 2

The graph of the system is the intersection of the shaded regions.

Graph each inequality in the system. Use different shades for each region.

Identify the region that is __common __to both graphs.