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.


exp-inequality-two-variables

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