# 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
$\le $
or
$\ge $
is used because the boundary is included in the solution. Use a dashed line if
$<\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\text{\hspace{0.17em}}>$
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
$\ge $
or
$>$
is used, shade above the line. If the symbol
$\le $
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 $y<2x+1$

**
Example 2:
**

Graph the system of linear inequalities.

${y}{<}\frac{{1}}{{3}}{x}{+}{4}$

${y}{\ge}{-}{3}{x}{-}{1}$

${y}{\ge}{5}{x}{+}{1}$

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