Browse Questions

# Solve the system of inequalities graphically $x+2y \leq 10 ; x+y \geq 1 ; x -y \leq 0 ; x \leq 0 , y\leq 0$

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.
• 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 $x+2y \leq 10$---------(1)
Consider the equation $x+2y =10$
The points $(10,0)$ and $(0,5)$ 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, $0+ 2(0) \leq 10$
$=> 0 \leq 10$ is true.
Hence the inequality (1) represents the region below the line $x+2y =10$ containing the point (0,0) (including the line )
Step 2:
The second inequality is $x+y \leq 1$ -----(2)
Consider the equation $x+y =1$
We see that $(1,0)$ and $(0,1)$ satisfy the equation .
The graph of the equation is drawn as shown.
The line divides the XY plane into two regions ( half plane)
Consider the point (0,0)
We see that $0+0 \geq 1$
$=> 0 \leq 1$ is false
Thus the inequality (2) is represented by the region above the line $x+y=1$ not including the point (0,0) (including the line y=2x)
Step 3 :
The third inequality is $x-y \leq 0$ -----(3)
Consider the equation $x-y =0$
We see that $(0,0) ,(1,1)$ Satisfy equation.
The graph of the equation is drawn as shown.
The line divides the XY plane into two regions ( half plane)
Consider the point (1,0)
We see that $1-0 \leq 0$
$1 \leq 0$ is false
Thus the inequality (3) is represented by the region above the line $x-y=0$ not including the point (0,0) (not including the line )
Step 4:
The fourth inequality is $x \geq 0, y \geq 0$
Step 5:
Hence the solution of the system of linear inequalities is represented by the common shaded region in the first quadrant including the line.