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 variables.

The points \left( {{\text{x}},{\text{y}}} \right) which satisfy the linear inequality in two variables, ‘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.
The solution set of inequality lies on the origin side.


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