# 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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