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Graphical Solution of Linear inequalities in Two Variable for Class 11 ,CBSE Board, IITJEE maths and other exams




Graphical Solution of Linear inequalities in Two Variable.

A linear equation in two variable is of the form
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

How to find the solution graphicaly for Linear inequalities in Two Variable.

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

ie, ax+by+c=0

2) Use a dashed or dotted line if the problem involves a strict inequality, < or >.
3) Otherwise, use a solid line to indicate that the line   itself constitutes part of the solution.
4) 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.
5) 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

Example

Determine the solution set for the inequality
x+y > 1

Solution

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

 

How to find the solution graphicaly for pair of  Linear inequalities in Two Variable.

ax+by+c< 0

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.

Steps
1) Find the graphical solution for each inequality independently using the technique decsribed above
2) Now determine the region in common with each solution set



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