# Solve the system of inequalities graphically $3x +2y \leq 12 ; x \geq 1;y \geq 2$

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:
Consider the inequality $3x +2y \leq 12$----(1)
Consider the corresponding equation $3x +2y =12$
The points $(4,0)$ and $(0,6)$ satisfy the equation.
The Graph of this line is shown in the diagram below.
Step 2:
The second inequality is $x \geq 1$ ----(2)
Consider $x=1$ The graph of this line is shown in the diagram.
The third inequality is $y \geq 2$
Step 3:
consider $y=2$ The graph if this line is shown in the diagram
The inequality (1) represented the half plane below the line $3x+2y =12$ (including the line)
The inequality (2) represented the half plane on the right of the line $x=1$ (including the line x=1)
The inequality (3) represented the half plane below 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.