Solve the system of inequalities graphically.
1. $x\geq 3$, $y\geq 2$
Consider, $x=3$
Consider origin (0,0)
$0\geq 3$ which is not true
The portion which does not contain (0,0), represents solution set of $x\geq 3$
Consider, $y=2$
Consider origin (0,0)
$0\geq 2$ which is not true
The portion which does not contain (0,0), represents solution set of $y\geq 2$
Thus, the green shaded region, common to the above two shaded regions, is the required solution region of the given system of inequalities.
2. $3x+2y\leq 12$, $x\geq1$, $y\geq2$
Consider, $3x+2y=12$
Put x=0, y=6
Put y=0, x=4
Consider origin (0,0)
$0\leq 12$ which is true
The portion which contain (0,0), represents solution set of $3x+2y\leq 12$
Consider, $x=1$
Consider origin (0,0)
$0\geq 1$ which is not true
The portion which does not contain (0,0), represents solution set of $x\geq 1$
Consider, $y=2$
Consider origin (0,0)
$0\geq 2$ which is not true
The portion which does not contain (0,0), represents solution set of $y\geq 2$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
3. $2x+y\geq 6$, $3x+4y\leq 12$
Consider, $2x+y=6$
Put x=0, y=6
Put y=0, x=3
Consider origin (0,0)
$0\geq 6$ which is not true
The portion which does not contain (0,0), represents solution set of $2x+y\geq 6$
Consider, $3x+4y=12$
Put x=0, y=3
Put y=0, x=4
Consider origin (0,0)
$0\leq 12$ which is true
The portion which contain (0,0), represents solution set of $3x+4y\leq 12$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
4. $x+y>4$, $2x-y>0$
Consider, $x+y=4$
Put x=0, y=4
Put y=0, x=4
Consider origin (0,0)
$0>4$ which is not true
The portion which does not contain (0,0), represents solution set of $x+y>4$
Consider, $2x-y=0$
Put x=0, y=0
Put x=1, y=2
Consider point (1,1)
$1>0$ which is true
The portion which contain (1,1), represents solution set of $2x-y>0$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
5. $2x-y>1$, $x-2y<-1$
Consider, $2x-y=1$
Put x=0, y=-1
Put y=0, x=1/2
Consider origin (0,0)
$0>1$ which is not true
The portion which does not contain (0,0), represents solution set of $2x-y>1$
Consider, $x-2y=-1$
Put x=0, y=1/2
Put y=0, x=-1
Consider origin (0,0)
$0<-1$ which is not true
The portion which does not contain (0,0), represents solution set of $x-2y<-1$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
6. $x+y\leq 6$, $x+y\geq 4$
Consider, $x+y=6$
Put x=0, y=6
Put y=0, x=6
Consider origin (0,0)
$0\leq 12$ which is true
The portion which contain (0,0), represents solution set of $x+y\leq 6$
Consider, $x+y=4$
Put x=0, y=4
Put y=0, x=4
Consider origin (0,0)
$0\geq 1$ which is not true
The portion which does not contain (0,0), represents solution set of $x+y\geq 4$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
7. $2x+y\geq 8$, $x+2y\geq 10$
Consider, $2x+y=8$
Put x=0, y=8
Put y=0, x=4
Consider origin (0,0)
$0\leq 12$ which is not true
The portion which does not contain (0,0), represents solution set of $2x+y\geq 8$
Consider, $x+2y=10$
Put x=0, y=5
Put y=0, x=10
Consider origin (0,0)
$0\geq 1$ which is not true
The portion which does not contain (0,0), represents solution set of $x+2y\geq 10$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
8. $x+y\leq 9$, $y>x$, $x\geq 0$
Consider, $x+y=9$
Put x=0, y=6
Put y=0, x=4
Consider origin (0,0)
$0\leq 9$ which is true
The portion which contain (0,0), represents solution set of $x+y\leq 9$
Consider, $y=x$
Put x=1, y=1
Put y=2, x=2
Consider point(0,1)
$1>0$ which is true
The portion which contain (0,1), represents solution set of $y>x$
Consider, $x\geq 0$
The portion on right side of y-axis, represents solution set of $x\geq 0$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
9. $5x+4y\leq 20$, $x\geq 1$, $y\geq 2$
Consider, $5x+4y=20$
Put x=0, y=5
Put y=0, x=4
Consider origin (0,0)
$0\leq 20$ which is true
The portion which contain (0,0), represents solution set of $5x+4y\leq 20$
Consider, $x=1$
Consider origin (0,0)
$0\geq 1$ which is not true
The portion which does not contain (0,0), represents solution set of $x\geq 1$
Consider, $y=2$
Consider origin (0,0)
$0\geq 2$ which is not true
The portion which does not contain (0,0), represents solution set of $y\geq 2$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
10. $3x+4y\leq 60$, $x+3y\leq 30$, $x\geq 0$, $y\geq 0$
Consider, $3x+4y=60$
Put x=0, y=15
Put y=0, x=20
Consider origin (0,0)
$0\leq 60$ which is true
The portion which contain (0,0), represents solution set of $3x+4y\leq 60$
Consider, $x+3y=30$
Put x=0, y=10
Put y=0, x=30
Consider origin (0,0)
$0\leq 30$ which is true
The portion which contain (0,0), represents solution set of $x+3y\leq 30$
Consider, $x\geq 0$
The portion on right side of y-axis, represents solution set of $x\geq 0$ Consider, $y\geq 0$
The portion above x-axis, represents solution set of $y\geq 0$ Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
11. $2x+y\geq 4$, $x+y\leq 3$ $2x-3y\leq 6$
Consider, $2x+y=4$
Put x=0, y=4
Put y=0, x=2
Consider origin (0,0)
$0\geq 4$ which is not true
The portion which does not contain (0,0), represents solution set of $2x+y\geq 4$
Consider, $x+y=3$
Put x=0, y=3
Put y=0, x=3
Consider origin (0,0)
$0\leq 3$ which is true
The portion which contain (0,0), represents solution set of $x+y\leq 3$
Consider, $2x-3y=6$
Put x=0, y=-2
Put y=0, x=3
Consider origin (0,0)
$0\leq 6$ which is true
The portion which contain (0,0), represents solution set of $2x-3y\leq 6$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
12. $x-2y\leq 3$, $3x+4y\geq 12$, $x\geq 0$, $y\geq 1$
Consider, $x-2y=3$
Put x=0, y=-3/2
Put y=0, x=3
Consider origin (0,0)
$0\leq 12$ which is true
The portion which contain (0,0), represents solution set of $x-2y\leq 3$
Consider, $3x+4y=12$
Put x=0, y=3
Put y=0, x=4
Consider origin (0,0)
$0\leq 12$ which is true
The portion which contain (0,0), represents solution set of $3x+4y\geq 12$
Consider, $x\geq 0$
The portion on right side of y-axis, represents solution set of $x\geq 0$
Consider, $y=1$
Consider origin (0,0)
$0\geq 1$ which is not true
The portion which does not contain (0,0), represents solution set of $y\geq 1$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
13. $4x+3y\leq 60$, $y\geq 2x$, $x\geq 3$, $x,y\geq 0$
Consider, $4x+3y=60$
Put x=0, y=20
Put y=0, x=15
Consider origin (0,0)
$0\leq 12$ which is true
The portion which contain (0,0), represents solution set of $4x+3y\leq 60$
Consider, $y=2x$
Put x=1, y=2
Put x=2, y=4
Consider point(0,1)
$1\geq 0$ which is true
The portion which contain (0,1), represents solution set of $y\geq 2x$
Consider, $x=3$
Consider origin (0,0)
$0\geq 1$ which is not true
The portion which does not contain (0,0), represents solution set of $x\geq 1$
Consider, $x,y\geq 0$
1st quadrant represents solution set of $x,y\geq 0$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
14. $3x+2y\leq150$, $x+4y\leq 80$, $x\leq 15$, $y\geq 0$
Consider, $3x+2y\leq150$
Put x=0, y=75
Put y=0, x=50
Consider origin (0,0)
$0\leq 150$ which is true
The portion which contain (0,0), represents solution set of $3x+2y\leq150$
Consider, $x+4y\leq 80$
Put x=0, y=20
Put y=0, x=80
Consider point(0,0)
$0\leq 80$ which is true
The portion which contain (0,0), represents solution set of $x+4y\leq 80$
Consider, $x=15$
Consider origin (0,0)
$0\leq 15$ which is true
The portion which contain (0,0), represents solution set of $x\leq 15$
Consider, $y\geq 0$
The portion above x-axis, represents solution set of $y\geq 0$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
15. $x+2y\leq 10$, $x+y\geq 1$, $x-y\leq 0$, $x\geq 0$, $y\geq 0$
Consider, $x+2y=10$
Put x=0, y=5
Put y=0, x=10
Consider origin (0,0)
$0\leq 10$ which is true
The portion which contain (0,0), represents solution set of $x+2y\leq 10$
Consider, $x+y=1$
Put x=0, y=1
Put y=0, x=1
Consider origin (0,0)
$0\geq 1$ which is not true
The portion which does not contain (0,0), represents solution set of $x+y\geq 1$
Consider, $x-y=0$
Put x=1, y=1
Put y=2, x=2
Consider point(0,1)
$-1\leq 0$ which is true
The portion which contain (0,1), represents solution set of $x-y\leq 0$
Consider, $x,y\geq 0$
1st quadrant represents solution set of $x,y\geq 0$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
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1. $x\geq 3$, $y\geq 2$
Consider, $x=3$
Consider origin (0,0)
$0\geq 3$ which is not true
The portion which does not contain (0,0), represents solution set of $x\geq 3$
Consider, $y=2$
Consider origin (0,0)
$0\geq 2$ which is not true
The portion which does not contain (0,0), represents solution set of $y\geq 2$
Thus, the green shaded region, common to the above two shaded regions, is the required solution region of the given system of inequalities.
2. $3x+2y\leq 12$, $x\geq1$, $y\geq2$
Consider, $3x+2y=12$
Put x=0, y=6
Put y=0, x=4
Consider origin (0,0)
$0\leq 12$ which is true
The portion which contain (0,0), represents solution set of $3x+2y\leq 12$
Consider, $x=1$
Consider origin (0,0)
$0\geq 1$ which is not true
The portion which does not contain (0,0), represents solution set of $x\geq 1$
Consider, $y=2$
Consider origin (0,0)
$0\geq 2$ which is not true
The portion which does not contain (0,0), represents solution set of $y\geq 2$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
3. $2x+y\geq 6$, $3x+4y\leq 12$
Consider, $2x+y=6$
Put x=0, y=6
Put y=0, x=3
Consider origin (0,0)
$0\geq 6$ which is not true
The portion which does not contain (0,0), represents solution set of $2x+y\geq 6$
Consider, $3x+4y=12$
Put x=0, y=3
Put y=0, x=4
Consider origin (0,0)
$0\leq 12$ which is true
The portion which contain (0,0), represents solution set of $3x+4y\leq 12$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
4. $x+y>4$, $2x-y>0$
Consider, $x+y=4$
Put x=0, y=4
Put y=0, x=4
Consider origin (0,0)
$0>4$ which is not true
The portion which does not contain (0,0), represents solution set of $x+y>4$
Consider, $2x-y=0$
Put x=0, y=0
Put x=1, y=2
Consider point (1,1)
$1>0$ which is true
The portion which contain (1,1), represents solution set of $2x-y>0$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
5. $2x-y>1$, $x-2y<-1$
Consider, $2x-y=1$
Put x=0, y=-1
Put y=0, x=1/2
Consider origin (0,0)
$0>1$ which is not true
The portion which does not contain (0,0), represents solution set of $2x-y>1$
Consider, $x-2y=-1$
Put x=0, y=1/2
Put y=0, x=-1
Consider origin (0,0)
$0<-1$ which is not true
The portion which does not contain (0,0), represents solution set of $x-2y<-1$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
6. $x+y\leq 6$, $x+y\geq 4$
Consider, $x+y=6$
Put x=0, y=6
Put y=0, x=6
Consider origin (0,0)
$0\leq 12$ which is true
The portion which contain (0,0), represents solution set of $x+y\leq 6$
Consider, $x+y=4$
Put x=0, y=4
Put y=0, x=4
Consider origin (0,0)
$0\geq 1$ which is not true
The portion which does not contain (0,0), represents solution set of $x+y\geq 4$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
7. $2x+y\geq 8$, $x+2y\geq 10$
Consider, $2x+y=8$
Put x=0, y=8
Put y=0, x=4
Consider origin (0,0)
$0\leq 12$ which is not true
The portion which does not contain (0,0), represents solution set of $2x+y\geq 8$
Consider, $x+2y=10$
Put x=0, y=5
Put y=0, x=10
Consider origin (0,0)
$0\geq 1$ which is not true
The portion which does not contain (0,0), represents solution set of $x+2y\geq 10$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
8. $x+y\leq 9$, $y>x$, $x\geq 0$
Consider, $x+y=9$
Put x=0, y=6
Put y=0, x=4
Consider origin (0,0)
$0\leq 9$ which is true
The portion which contain (0,0), represents solution set of $x+y\leq 9$
Consider, $y=x$
Put x=1, y=1
Put y=2, x=2
Consider point(0,1)
$1>0$ which is true
The portion which contain (0,1), represents solution set of $y>x$
Consider, $x\geq 0$
The portion on right side of y-axis, represents solution set of $x\geq 0$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
9. $5x+4y\leq 20$, $x\geq 1$, $y\geq 2$
Consider, $5x+4y=20$
Put x=0, y=5
Put y=0, x=4
Consider origin (0,0)
$0\leq 20$ which is true
The portion which contain (0,0), represents solution set of $5x+4y\leq 20$
Consider, $x=1$
Consider origin (0,0)
$0\geq 1$ which is not true
The portion which does not contain (0,0), represents solution set of $x\geq 1$
Consider, $y=2$
Consider origin (0,0)
$0\geq 2$ which is not true
The portion which does not contain (0,0), represents solution set of $y\geq 2$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
10. $3x+4y\leq 60$, $x+3y\leq 30$, $x\geq 0$, $y\geq 0$
Consider, $3x+4y=60$
Put x=0, y=15
Put y=0, x=20
Consider origin (0,0)
$0\leq 60$ which is true
The portion which contain (0,0), represents solution set of $3x+4y\leq 60$
Consider, $x+3y=30$
Put x=0, y=10
Put y=0, x=30
Consider origin (0,0)
$0\leq 30$ which is true
The portion which contain (0,0), represents solution set of $x+3y\leq 30$
Consider, $x\geq 0$
The portion on right side of y-axis, represents solution set of $x\geq 0$ Consider, $y\geq 0$
The portion above x-axis, represents solution set of $y\geq 0$ Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
11. $2x+y\geq 4$, $x+y\leq 3$ $2x-3y\leq 6$
Consider, $2x+y=4$
Put x=0, y=4
Put y=0, x=2
Consider origin (0,0)
$0\geq 4$ which is not true
The portion which does not contain (0,0), represents solution set of $2x+y\geq 4$
Consider, $x+y=3$
Put x=0, y=3
Put y=0, x=3
Consider origin (0,0)
$0\leq 3$ which is true
The portion which contain (0,0), represents solution set of $x+y\leq 3$
Consider, $2x-3y=6$
Put x=0, y=-2
Put y=0, x=3
Consider origin (0,0)
$0\leq 6$ which is true
The portion which contain (0,0), represents solution set of $2x-3y\leq 6$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
12. $x-2y\leq 3$, $3x+4y\geq 12$, $x\geq 0$, $y\geq 1$
Consider, $x-2y=3$
Put x=0, y=-3/2
Put y=0, x=3
Consider origin (0,0)
$0\leq 12$ which is true
The portion which contain (0,0), represents solution set of $x-2y\leq 3$
Consider, $3x+4y=12$
Put x=0, y=3
Put y=0, x=4
Consider origin (0,0)
$0\leq 12$ which is true
The portion which contain (0,0), represents solution set of $3x+4y\geq 12$
Consider, $x\geq 0$
The portion on right side of y-axis, represents solution set of $x\geq 0$
Consider, $y=1$
Consider origin (0,0)
$0\geq 1$ which is not true
The portion which does not contain (0,0), represents solution set of $y\geq 1$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
13. $4x+3y\leq 60$, $y\geq 2x$, $x\geq 3$, $x,y\geq 0$
Consider, $4x+3y=60$
Put x=0, y=20
Put y=0, x=15
Consider origin (0,0)
$0\leq 12$ which is true
The portion which contain (0,0), represents solution set of $4x+3y\leq 60$
Consider, $y=2x$
Put x=1, y=2
Put x=2, y=4
Consider point(0,1)
$1\geq 0$ which is true
The portion which contain (0,1), represents solution set of $y\geq 2x$
Consider, $x=3$
Consider origin (0,0)
$0\geq 1$ which is not true
The portion which does not contain (0,0), represents solution set of $x\geq 1$
Consider, $x,y\geq 0$
1st quadrant represents solution set of $x,y\geq 0$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
14. $3x+2y\leq150$, $x+4y\leq 80$, $x\leq 15$, $y\geq 0$
Consider, $3x+2y\leq150$
Put x=0, y=75
Put y=0, x=50
Consider origin (0,0)
$0\leq 150$ which is true
The portion which contain (0,0), represents solution set of $3x+2y\leq150$
Consider, $x+4y\leq 80$
Put x=0, y=20
Put y=0, x=80
Consider point(0,0)
$0\leq 80$ which is true
The portion which contain (0,0), represents solution set of $x+4y\leq 80$
Consider, $x=15$
Consider origin (0,0)
$0\leq 15$ which is true
The portion which contain (0,0), represents solution set of $x\leq 15$
Consider, $y\geq 0$
The portion above x-axis, represents solution set of $y\geq 0$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
15. $x+2y\leq 10$, $x+y\geq 1$, $x-y\leq 0$, $x\geq 0$, $y\geq 0$
Consider, $x+2y=10$
Put x=0, y=5
Put y=0, x=10
Consider origin (0,0)
$0\leq 10$ which is true
The portion which contain (0,0), represents solution set of $x+2y\leq 10$
Consider, $x+y=1$
Put x=0, y=1
Put y=0, x=1
Consider origin (0,0)
$0\geq 1$ which is not true
The portion which does not contain (0,0), represents solution set of $x+y\geq 1$
Consider, $x-y=0$
Put x=1, y=1
Put y=2, x=2
Consider point(0,1)
$-1\leq 0$ which is true
The portion which contain (0,1), represents solution set of $x-y\leq 0$
Consider, $x,y\geq 0$
1st quadrant represents solution set of $x,y\geq 0$
Thus, the shaded region, common to the above inequalities, is the required solution of the given system of inequalities.
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