Examples of Linear Inequalities in Two Variables

Example:
Graph the solution set of the system of linear inequalities

\begin{gathered} 3{\text{x}} + 5{\text{y}} \leqslant 9 \\ {\text{x}} - 6{\text{y}} \leqslant 3 \\ \end{gathered}

Solution:

We have

\begin{gathered} 3{\text{x}} + 5{\text{y}} \leqslant 9\,\,\, - - - \left( A \right) \\ {\text{x}} - 6{\text{y}} \leqslant 3\,\,\, - - - \left( B \right) \\ \end{gathered}

The corresponding equations of inequalities (A) and (B)

3{\text{x}} + 5{\text{y}} = 9\,\,\, - - - \left( 1 \right) {\text{x}} - 6{\text{y}} = 3\,\,\, - - - \left( 2 \right)
For x – intercept For x – intercept
Put {\text{y}} = 0 in (1) we get Put {\text{y}} = 0 in (2) we get
 \Rightarrow 3{\text{x}} + 5\left( {\text{0}} \right) = 9  \Rightarrow {\text{x}} - 6\left( {\text{0}} \right) = 3
 \Rightarrow 3{\text{x}} = 9  \Rightarrow {\text{x}} = 3
 \Rightarrow {\text{x}} = 3
\therefore \left( {3,0} \right) \therefore \left( {3,0} \right)
For y – intercept For y – intercept
Put {\text{x}} = 0 in (1) we get Put {\text{x}} = 0 in (2) we get
 \Rightarrow 3\left( {\text{0}} \right) + 5{\text{y}} = 9  \Rightarrow {\text{0}} - 6{\text{y}} = 3
 \Rightarrow 5{\text{y}} = 9  \Rightarrow  - 6{\text{y}} = 3
 \Rightarrow {\text{y}} = \frac{9}{5}  \Rightarrow {\text{y}} = - \frac{1}{2}
\therefore \left( {0,\frac{9}{5}} \right) \therefore \left( {0, - \frac{1}{2}} \right)
Test Test
Put origin \left( {0,0} \right) in eq (A) Put origin \left( {0,0} \right) in eq (B)
0 + 0 < 9 0 - 0 < 3
 \Rightarrow 0 < 9 which is true solution set origin side.  \Rightarrow 0 < 3 which is true solution set origin side.


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