$ax+by+c=0$

We have already studied in Coordinate geometry that this can be represented by a straight line in x-y plane. All the points on the straight line are the solutions of this linear equation.

we can similarly find the solution set graphically for the linear inequalities in the below form

$ax+by+c< 0$

$ax+by+c> 0$

$ax+by+c \geq 0$

$ax+by+c \leq 0 $

- Draw the graph of the equation obtained for the given inequality by replacing the inequality sign with an equal sign.

$ax+by+c=0 $

- This can be done easily by Point on the x-axis( x,0) and point on the y axis ( 0,y)

- Point on x-axis given by $ax+b(0)+c=0 $ or $x=\frac {-c}{a}$ or $(\frac {-c}{a} ,0)$

- Point on y-axis given by $a(0)+by+c=0 $ or $y=\frac {-c}{b}$ or $(0,\frac {-c}{b})$

- Locate these point on cartesian plane and join them to find the line

- This can be done easily by Point on the x-axis( x,0) and point on the y axis ( 0,y)
- Use a dashed or dotted line if the problem involves a strict inequality, < or >.
- Otherwise, use a solid line to indicate that the line itself constitutes part of the solution.
- Pick a point lying in one of the half-planes determined by the line sketched in step 1 and substitute the values of x and y into the given inequality.

Use the origin whenever possible. - If the inequality is satisfied, the graph of the inequality includes the half-plane containing the test point.

Otherwise, the solution includes the half-plane not containing the test point

Determine the solution set for the inequality

$x+y > 1$

1) Draw the graph of the equation obtained for the given inequality by replacing the inequality sign with an equal sign.

i.e $x+y=1 $

2) Pick the test point as origin (0,0), and put into the inequality

$0+0 > 1$

$0> 1$

Which is false

So the solution set is other half plane of the line

$px+qy+c < 0$

The solution set of a system of linear inequalities in two variables x and y is the set of all points (x, y) that satisfy each inequality of the system.

- Find the graphical solution for each inequality independently using the technique decsribed above
- Now determine the region in common with each solution set

Find the solution of the below system of inequalities

$2x+3y >1$

$x+2y >2$

$x>1$

1) for $2x+3y >1$, Solving using the above method solution is

2)for $x+2y >2$, Solving using the above method solution is

3) for $x>1$, Solving using the above method solution is

4) Now we draw these on the single graph and can determine the common region

**Notes**- What are inequalities
- Things which changes the direction of the inequality
- Linear Inequation in One Variable
- Linear Inequation in Two Variable
- Steps to solve the inequalities in one variable
- Steps to solve the inequality of the another form
- Quadratic Inequation
- Steps to solve Quadratic or polynomial inequalities
- Cubic Inequation
- Steps to solve Cubic inequalities
- Absolute value equation
- Absolute value inequation
- Graphical Solution of Linear inequalities in Two Variable

**NCERT Solutions**