A linear inequality represents an area of the coordinate plane that has a boundary line. Every point in that region is represents the solution of the inequality. Simply a linear inequality is just everything on one side of a line on a graph.

To solve an inequality with the help of a graph, we have to follow some steps:

(1) First convert the inequality sign in to the equal sign.

(2) Draw the graph of the given equation. If the original problem(inequality)

contains the symbol "<" or ">", then we use a dashed line to draw the graph

and if the original inequality contains the symbol "$\leq$" or "$\geq$", then we

use a solid line to draw the graph.

(3) Noe select an arbitrary test point(not on the line), for convenience normally we

select (0,0) as the test point and substitute this test point in to the given

inequality,

(a) If this test point satisfies the inequality then shade the region on the side

of the line containing this point.

(b) If this test point does not satisfies the inequality then shade the region

on the side of the line not containing this point.

**Example 1:** Graph the linear inequality y $\leq$ 3x + 1.

**Solution:** For graph, we set the problem as y = 3x + 1

Pick a test point (0,0). This test point satisfies the given inequality,

as 0 $\leq$ 0 + 1

or 0 $\leq$ 1 which is true. So we have to shade the region on the side of

the line containing this point. The line is solid line since the given problem

is $\leq$.

**Example 2:** Graph y > $\frac{x}{3}$ + 1

**Solution:** First convert ">" is "=", then

y = $\frac{x}{3}$ + 1. Here m(slop) is 1/3. Draw a graph and select a test point (0,0). Now the test point does not satisfies the given inequality, so shade the region on the side of the line not containing this point. We draw a dashed line since the given inequality is ">".

**Example 3:** Graph the solution of system of inequalities.

2x + y $\leq$ 10

x -3y $\geq$ 6

**Solution:** To graph the solution, we follow below steps:

(1) Graph each inequality on the same set of axes.

(2) Check where of the shading of the inequalities overlaps

The overlapping region is the solution of the given set of inequalities.