# Solve the following system of inequalities graphically : $5x+4y \leq 20; x \geq 1; y \geq 2$

Toolbox:
• To represent the solution of linear inequality of one or two variable in a plane if the inequality involves $'\geq'$ or $' \leq$ we draw the graph of the line as a thick line to indicate the line is included in the solution set.
• If the inequality involves $'>'$ as $'<'$ we draw the graph of the line using is not included in the solution set.
• To solve an inequality $ax+by > c \qquad a \neq 0, b \neq 0 ( or \;> )$
• We consider the corresponding equation $ax+by =c$ which represents a straight line This line divides the plane into two half planes I and II
• We take any point in I half plane and check if it satisfies the given inequality will be one half plane (called solution region ) Containing the point satisfying the inequality
Step 1:
The first inequality is $5x+4y \leq 20$ ----------(1)
Consider the equation $5x+4y =20$
The point $(4,0)$ and $(0,5)$ satisfy the equation.
The graph of this line is drawn as shown.
The line divides the xy - plane into two half planes .
Consider the point (0,0)
We see that $5(0) +4(0) \leq 20$
$0 \leq 20$ is true.
Thus the region inequality represents the region below the line $5x+4y=20$ containing the point (0,0) (including the line)
Step 2:
The second inequality is $x \geq 1$ ---(2)
Consider the $x=1$ . The graph of the line is as shown.
The inequality represents the region on the right hand side of the line $x=1$ (including the line x=1)
Step 3:
The third inequality is $y \geq 2$
Consider y=2 . The graph is as shown.
The inequality represents the region above the line $y=2$ (including the line y=2)
Step 4:
Hence, the solution of the given system of linear inequalities is represented by the common shaded region including the points on the lines.