To plot the line of the inequality , we begin by inserting the values x = 0 and y = 0 one after the other.
[tex]\begin{gathered} 4x+5y>20 \\ \text{When x = 0, then} \\ 4(0)+5y=20 \\ 5y=20 \\ y=4 \\ \text{That gives us the point (0, 4)} \\ \text{Similarly when y = 0, then} \\ 4x+5(0)=20 \\ 4x=20 \\ x=5 \\ \text{That gives us the point (5, 0)} \\ \text{With these two points we can now plot a line by joining both points with a ruler} \\ \text{Note also that the inequality can now be expressed as;} \\ 4x+5y>20 \\ 5y>20-4x \\ y>\frac{20-4x}{5} \\ y>\frac{20}{5}-\frac{4x}{5} \\ y>4-\frac{4}{5}x \\ y>-\frac{4}{5}x+4 \end{gathered}[/tex]
Letus now derive two points for the second inequality given just like we did for the first one above;
[tex]\begin{gathered} -3x+7y\ge7 \\ \text{When x = 0, then } \\ -3(0)+7y=7 \\ 7y=7 \\ y=1 \\ \text{This gives us the point (0, 1)} \\ \text{Also, when y = 0, then} \\ -3x+7(0)=7 \\ -3x=7 \\ x=-\frac{7}{3}\text{ (OR x=-2}\frac{1}{3}) \\ \text{That gives us the point (-2}\frac{1}{3},0) \\ \text{The equation can now be expressed as} \\ -3x+7y\ge7 \\ 7y\ge7+3x \\ y\ge\frac{7+3x}{7} \\ y\ge\frac{7}{7}+\frac{3x}{7} \\ y\ge1+\frac{3}{7}x \\ y\ge\frac{3}{7}x+1 \end{gathered}[/tex]
The graph of both inequalities is now shown below;
Observe carefully that the solution lies in the region shaded BLUE and RED.
One point in this region is (4, 4)
The blue and red graphs are represented by;
[tex]\begin{gathered} y>-\frac{4}{5}x+4\text{ (RED graph)} \\ y\ge1+\frac{3}{7}x\text{ (BLUE graph)} \end{gathered}[/tex]
