Linear Inequalities in Two Variables

The inequalities of the form {\text{ax}} + {\text{by}} >  {\text{c}}, {\text{ax}} +  {\text{by}} \leqslant {\text{c}}, {\text{ax}}  + {\text{by}} \geqslant {\text{c}}, where {\text{a}} \ne 0, {\text{b}} \ne 0, c are constants, are called the linear inequalities in two variable.
The points \left( {{\text{x}},{\text{y}}} \right)which satisfy the linear inequality in two variable ‘x’ and ‘y’ from its solution.

Graphing the Solution Region of Linear Inequality in Two Variables:
Example:
Graph the solution set of the linear inequality {\text{x}}  - {\text{y}} \leqslant 1 in xy–plane.
Solution:
We have

{\text{x}} - {\text{y}} \leqslant 1\,\,\,  -  -   - \left( A \right)


The corresponding equation of inequality A

{\text{x}} - {\text{y}} = 1\,\,\,  -  -   - \left( i \right)


For x–intercept:
Put {\text{y}} = 0 in equation (i)

\begin{gathered} {\text{x}} - \left( {\text{0}} \right) = 1 \\ \Rightarrow {\text{x}} = 1 \\ \therefore \left( {1,0} \right) \\ \end{gathered}


For y–intercept:
Put {\text{x}} = 0 in equation (i)

\begin{gathered} {\text{0}} - {\text{y}} = 1 \\ \Rightarrow {\text{y}} =  - 1 \\ \therefore \left( {0, - 1} \right) \\ \end{gathered}


Test:
Put origin \left( {0,0} \right) in equation (A)

\begin{gathered} 0 - 0 < 1 \\ \Rightarrow 0 < 1 \\ \end{gathered}


Which is true.
Solution set of inequality lies origin side.


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