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Systems of Linear Inequalities

One linear inequality in two variables divides the plane into two half-planes . To graph the inequality, graph the equation of the boundary.  Use a solid line if the symbol or is used because the boundary is included in the solution.  Use a dashed line if < or > is used to indicate that the boundary is not part of the solution. Shade the appropriate region. Unless you are graphing a vertical line the sign of the inequality will let you know which half-plane to shade.  If the symbol or > is used, shade above the line.  If the symbol or < is used shade below the line. For a vertical line, larger solutions are to the right and smaller solutions are to the left. A system of two or more linear inequalities can divide the plane into more complex shapes.

Example 1:

Graph y < 2 x + 1

Math diagram

Example 2:

Graph the system of linear inequalities.

y < 1 3 x + 4

y 3 x 1

y 5 x + 1


Graphing the three lines and shading the region enclosed, we get the figure below.

Math diagram


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