Graphing the Solution Region of System of Linear Inequalities

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

\begin{gathered} 4{\text{x}} - 5{\text{y}} \leqslant 20 \\ 3{\text{x}} + 2{\text{y}} \leqslant 6 \\ 3{\text{x}} + {\text{y}} \leqslant 2 \\ \end{gathered}


Solution:
We have

\begin{gathered} 4{\text{x}} - 5{\text{y}} \leqslant 20\,\,\,  -  -   - \left( A \right) \\ 3{\text{x}} + 2{\text{y}} \leqslant 6\,\,\,\,  -  -   - \left( B \right) \\ 3{\text{x}} + {\text{y}} \leqslant 2\,\,\,\,  -  -   - \left( C \right) \\ \end{gathered}


The corresponding equations of inequalities (A), (B) and (C), we get

\begin{gathered} 4{\text{x}} - 5{\text{y}} = 20\,\,\, -  -  -  \left( 1 \right) \\ 3{\text{x}} + 2{\text{y}} = 6\,\,\,\, -  -  -  \left( 2 \right) \\ 3{\text{x}} + {\text{y}} = 2\,\,\,\, -  -  -  \left( 3 \right) \\ \end{gathered}


For x–Intercepts:
Put {\text{y}} = 0in Eq (1), Eq (2) and Eq (3) we get

 \Rightarrow 4{\text{x}} - 5\left( {\text{0}}    \right) = 20

 \Rightarrow 3{\text{x}} + 2\left( {\text{0}}    \right) = 6

    \Rightarrow 3{\text{x}} + {\text{0}} = 2

 \Rightarrow 4{\text{x}} = 20

 \Rightarrow 3{\text{x}} = 6

    \Rightarrow 3{\text{x}} = 2

 \Rightarrow {\text{x}} = 5

 \Rightarrow {\text{x}} = 2

    \Rightarrow {\text{x}} = \frac{2}{3}

\therefore \left( {5,0} \right)

\therefore \left( {2,0} \right)

\therefore    \left( {\frac{2}{3},0} \right)

For y–Intercepts:
Put {\text{x}} = 0in Eq (1), Eq (2) and Eq (3) we get

 \Rightarrow 4\left( {\text{0}} \right) -    5{\text{y}} = 20

 \Rightarrow 3\left( {\text{0}} \right) +    2{\text{y}} = 6

    \Rightarrow 3\left( {\text{0}} \right) + {\text{y}} = 2

 \Rightarrow     - 5{\text{y}} = 20

 \Rightarrow 2{\text{y}} = 6

    \Rightarrow {\text{y}} = 2

 \Rightarrow {\text{y}} =  - 4

 \Rightarrow {\text{y}} = 3

    \Rightarrow {\text{y}} = 2

\therefore \left( {0, - 4} \right)

\therefore \left( {0,3} \right)

\therefore    \left( {0,2} \right)

Test:
Put origin as test point in inequalities (A), (B) and (C).

0 - 0 < 20

 \Rightarrow 0 + 0 < 6

    \Rightarrow 0 + {\text{0}} < 2

 \Rightarrow 0 < 20 (True)

 \Rightarrow 0 < 6 (True)

    \Rightarrow 0 < 2 (True)

\therefore Solution Set towards origin side.

\therefore Solution Set towards origin side.

\therefore    Solution Set towards origin side.


exp-inequality-three-variables

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