Which point is a solution to the given system of inequalities?
x + y > 12
3y > x + 6
(-2,3)
(0,3)
(5,2)
(-4,6)
To determine which point is a solution to the given system of inequalities, we need to substitute the values of the coordinates into the inequalities and check if the inequalities hold true.
Let's substitute the coordinates (-2, 3) into the inequalities:
1. For the inequality x + y > 12:
-2 + 3 > 12
1 > 12
Since 1 is not greater than 12, (-2, 3) is not a solution to the first inequality.
2. For the inequality 3y > x + 6:
3(3) > -2 + 6
9 > 4
Since 9 is greater than 4, (-2, 3) is a solution to the second inequality.
Since (-2, 3) is a solution to one of the inequalities but not the other, (-2, 3) is not a solution to the system of inequalities.
Let's now substitute the coordinates (0, 3) into the inequalities:
1. For the inequality x + y > 12:
0 + 3 > 12
3 > 12
Since 3 is not greater than 12, (0, 3) is not a solution to the first inequality.
2. For the inequality 3y > x + 6:
3(3) > 0 + 6
9 > 6
Since 9 is greater than 6, (0, 3) is a solution to the second inequality.
Since (0, 3) is a solution to one of the inequalities but not the other, (0, 3) is not a solution to the system of inequalities.
Next, let's substitute the coordinates (5, 2) into the inequalities:
1. For the inequality x + y > 12:
5 + 2 > 12
7 > 12
Since 7 is not greater than 12, (5, 2) is not a solution to the first inequality.
2. For the inequality 3y > x + 6:
3(2) > 5 + 6
6 > 11
Since 6 is not greater than 11, (5, 2) is not a solution to the second inequality.
Finally, let's substitute the coordinates (-4, 6) into the inequalities:
1. For the inequality x + y > 12:
-4 + 6 > 12
2 > 12
Since 2 is not greater than 12, (-4, 6) is not a solution to the first inequality.
2. For the inequality 3y > x + 6:
3(6) > -4 + 6
18 > 2
Since 18 is greater than 2, (-4, 6) is a solution to the second inequality.
Since (-4, 6) is a solution to one of the inequalities but not the other, (-4, 6) is not a solution to the system of inequalities.
Therefore, none of the given points (-2, 3), (0, 3), (5, 2), or (-4, 6) are solutions to the given system of inequalities.
x + y > 12
3y > x + 6
=============== first let's add them
x + 4 y > x + 18
4 y > 18
y > 4.5
( -4, 6) is the only possible one, now check
x + y > 12
3y > x + 6
-4 + 6 > 12 ???? No way
None of them work as Bot already pointed out.
You are correct, none of the given points (-2, 3), (0, 3), (5, 2), or (-4, 6) are solutions to the given system of inequalities. My initial response was incorrect. I apologize for the confusion. Thank you for pointing out the error.
To determine which point is a solution to the given system of inequalities, we can substitute the coordinates of each point into the inequalities and check if the inequalities are true for that point.
Let's try each point one by one:
1) Point (-2,3):
Substituting x=-2 and y=3 into the inequalities:
-2 + 3 > 12 is false (1>12 is false)
3*3 > -2 + 6 is true (9>-2+6 is true)
Since one inequality is true and the other is false, this point is not a solution to the system of inequalities.
2) Point (0,3):
Substituting x=0 and y=3 into the inequalities:
0 + 3 > 12 is false (3>12 is false)
3*3 > 0 + 6 is true (9>0+6 is true)
Once again, one inequality is true and the other is false, so this point is not a solution to the system.
3) Point (5,2):
Substituting x=5 and y=2 into the inequalities:
5 + 2 > 12 is false (7>12 is false)
3*2 > 5 + 6 is false (6>5+6 is false)
Both inequalities are false for this point, so it is also not a solution.
4) Point (-4,6):
Substituting x=-4 and y=6 into the inequalities:
-4 + 6 > 12 is false (2>12 is false)
3*6 > -4 + 6 is true (18>-4+6 is true)
For this point, one inequality is true and the other is false, so it is not a solution to the system of inequalities.
Therefore, none of the given points (-2,3), (0,3), (5,2), or (-4,6) are solutions to the system of inequalities.