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 the true solution of set origin side. | $$ \Rightarrow 0 < 3$$ which is the true solution of set origin side. |
