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# Solve the system of linear inequalities graphically $2x +y \geq 4; x+y \leq 3; 2x-3y \leq 6$

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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
The first inequality is $2x+ y \geq 4$ ------(1)
Consider the equation $2x+y =4$
The points $(2,0)$ and $(0,4)$ satisfy the equation .
The graph of this line is drawn as shown.
The line divides the x-y plane into two half planes.
Consider the point $(0,0)$
We see that , $2(0)+0 \geq 4$
$0 \geq 4$ is false
Thus the inequality (1) represent the region above the line $2x+y=4$ containing the point (0,0) including the line
Step 2:
The second inequality is $x+y \leq 3$ ----(2)
Consider the equation $x+y=3$
The point (3,0) and (0,3) 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 $0+ 0 \leq 3$ is true.
Thus the inequality (2) represents the region below the line $x+y =3$ containing the point (0,0) (including the line)
Step 3:
the third inequality is $2x-3y \leq 6$
Consider the equation $2x-3y=6$
The point (3,0) and (0,-2) satisfy the equation.
The graph of the equation $2x-3y =6$ is drawn as shown.
The line divides the xy plane into two half planes.
Consider the point (0,0) we see that $2(0)-3( 0) \leq 6=>0 \leq 6$ is true.
Thus the inequality (3) represents the region above the line $2x-3y =6$ containing the point (0,0) (including the line 2x-3y=6)
Step 4:
Hence the solution of the given system of linear inequalities is represented by the common the lines.
answered Jul 31, 2014 by