 Systems of linear inequalities are sets of two or more linear inequalities involving two or more variables.  Remember, the highest power of any variable in a linear inequality is 1.  For the system of linear inequalities to be satisfied, each linear inequality must be satisfied.  The most common kind of system of linear inequalities has two variables and two inequalities, but they can have any number.

 There are three main steps in graphing a system of linear inequalities. 1.Put each inequality in a useful form. 2.Replace each inequality with an equality and graph the lines that represent these equalities. 3.Find and shade the region where every inequality is satisfied.

 Graph the following system of linear inequalities: x > 1 2x + y ≤ 8 y ≥ 2 + x

 We want to get the inequalities into a form that makes them easy to graph.  We’ll use slope-intercept form.  We’re left with: x > 1 y ≤ -2x + 8 y ≥ x + 2

 Now we want to replace each inequality with an equality and graph the lines that represent these equalities. x = 1 y = -2x + 8 y = x + 2  If we have a strictly greater than or strictly less than inequality, we use a dotted line to represent it. Otherwise, we use a solid line.  This means that our first line will be dotted, and the others will be solid.

This is the result.

 Now you need to determine which side of each line fulfills the inequality that corresponds to that line.  In our graph, our region must be below the blue line, above the purple line, and to the right of the red line.  If it helps, draw arrows on each line to visualize which side is the solution to its inequality.

Here’s our graph again, marked with arrows to show where our solution set is with respect to each line.

Now, we shade in the solution region, which is the region that satisfies all of the inequalities: all of the arrows point into it. Here it happens to be the small triangle in the center of the graph.