Browse Questions

# Solve the following system of inequalities graphically $2x+y \geq 8; x+2y \geq 10$

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 $2x+y \geq 8$
Consider the equation $2x +y =8$
The points $(4,0) ,(0,8)$ satisfy the equation.
The graph of this is drawn as shown.
The line divides the xy plane into two half planes .
Consider the point $(0,0)$
We see $2(0) +0 \geq 8$
$0 \geq 8$ is false
Thus the inequality $2x+8 \geq 8$ represents the region above the line not containing the point (0,0) (including the line 2x+y=8)
Step 2:
The second inequality is $x+2y \leq 10$ ---------(2)
Consider the equation : $x+2y =10$
The points $(10,0)$ 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)
$0+2(0) \geq 10$ is false
Thus the inequality $x+2y \geq 10$ represents the region above the line not containing the point (0,0).
including the line $x+2y=10$
Step 3:
Hence the solution of the given system of linear inequalities is represented by the common shaded region, including the points on the lines.