In mathematics, any algebraic expression whose degree is one is known as linear expression. If any inequality holds a linear relation between the variables then such type of inequality is called as linear inequality. Like:
2x + y < 7, x - 2y + z > 4 etc, in these examples all the variables are presents in linear form, hence these are the linear inequalities.

## Systems of Linear Inequalities

A system of linear inequalities is a set of linear inequalities that we deal with all at once in the same plane. Usually a system of linear inequalities is just two or more than two inequalities together.
Generally a system of linear inequalities express as:
a11 x1 + a12 x2 +.........+ a1n xn $\leq or = or \geq$ b1
a21 x1 + a22 x2 +.........+ a2n xn $\leq or = or \geq$ b2
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
am1 x1 + am2 x2 +.........+ amn xn $\leq or = or \geq$ bm
Here x1 , x2,.....,xn are the unknown variables, a11, a12,.....,amn are the coefficients and b1, b2.....bm are the constants. This can be written as
AX $\leq or = or \geq$ B.

## Linear Inequalities in Two variables

If an inequality holds a linear relation between two variables then that type of inequality is called linear inequality of two variables. These variables are presents in the linear form in the inequality.
Like:
a11x1 + a12x2 $\leq or = or \geq$ b1 is the example of linear inequality of two variables.
Examples:
(a) 2a + 3b $\leq$ 7
(b) 3x - y > 9.

## Solving Linear Inequalities

Solving linear inequality is the same as solving linear equation. If x+1<6 is the given inequality the values can be positive and negative which satisfied the given inequality. There are many solutions for inequalities. It is easy to solve an inequality with one variable.

Example 1: Solve 3x - 4 < 5.
Solution: We have 3x - 4 < 5
3x -4 + 4 < 5 + 4
or 3x < 9
Divided both sides by 3
$\frac{3x}{3} <$\frac{9}{3}
or x < 3.

Example 2: Solve 6 - 4x $\leq$ 21 + x
Solution: Given inequality 6 - 4x $\leq$ 21 + x
Subtract 6 and x from both sides
6 - 6 - 4x -x $\leq$ 21 + x -6 -x
or -5x $\leq$ 15
Divide both sides by -5 and this time we change the direction of the inequality
x $\geq$ -3
So x can be equal to -3 and greater then -3.

## Graphing Linear Inequalities

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:
(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.