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# Solve the system of inequalities graphically $4x+3y \leq 60 ; y \geq 2x; x \geq 3 ; x,y \geq 0$

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 $4x +3y \leq 60$
Consider the equation $4x +3y =60$
The points $(15,0)$ and $(0,20)$ satisfy the equation .
The graph of the equation is drawn as shown.
The line divides the xy plane into two half planes.
Consider the point (0,0)
We see that, $4(0) +3(0) \leq 60$
=> $0 \leq 60$ is true.
Thus the inequality (1) is represented by the region below the line $4x+3y=60$ containing the point (0,0) (including the line )
Step 2 :
The second inequality is $y \geq 2x$ -------(2)
Consider the equation $y=2x$
The points $(0,0) ,(2,4)$ satisfy the equation.
The graph of the line is drawn as shown.
The line divides the XY plane into two half planes.
Consider the point (1,0)
We see that, $0 \geq 2(1)$
$0 \geq 2$ is false.
Thus the inequality (2) is represented by the region above the line $y=2x$ not including the point (1,0) (including the line y=2x)
Step 3 :
The third inequality is $x \geq 3$ -------(3)
The inequality represents the region to the right of the line $x=3$ (including the line)
Step 4 :
The 4th inequality is $x,y \geq 0$
It represent the region in the first quadrant of the XY -plane including positive x-axis and positive y-axis.
Step 5:
Hence the solution of the system of linear inequalities is represented by the common shaded region including the points on the lines and y-axis.
edited Aug 1, 2014 by meena.p