Browse Questions

# Solve the system of inequalities graphically $x-2y \leq 3; 3x +4y \geq 12 ; x \geq 0 ; y \geq 1$

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 $x-2y \leq 3$
Consider the equation $x-2y =3$
The points $(3,0)$ and $(0,-3/2)$
The graph of the line is drawn as shown.
The line divides the XY plane into two half planes.
Consider the point $(0,0)$
We see that $0-2(0) \leq 3$
=> $0 \leq 3$ is true.
Thus the inequality $x-2y \leq 3$ is represented by the region above the line $x-2y =3$ containing the point (0,0) (including the line)
Step 2:
The second inequality is $3x+4y \geq 12$------(2)
Consider the equation $3x+4y=12$
we see that the points (4,0) and (0,3) 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 $3(0) +4(0) \geq 12$
=> $0 \geq 12$ is false.
Thus the inequality (2) is represented by the region below the line $3x+4y=12$ containing the point (0,0)
Step 3:
The third inequality is $x \leq 0$ -----(3)
It is represented by the region to the right of y axis including the y- axis.
Step 4:
The 4th inequality is $y \geq 1$---------(4)
Consider equation $y=1$
The graph is drawn as shown . The inequality (4) represents the region above the line $y=1$ (including the line y=1)
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.