Question 1:
We have the region given by:
[tex]y> 2x + 1[/tex]
First we want to find an ordered pair that is not a solution, that is, evaluate the inequality in a pair (x, y) and that it is not fulfilled.
Example:
[tex](x, y) = (3,1)[/tex]
We replace:
[tex]1> 2 (3) +1\\1> 6 + 1\\1> 7[/tex]
It is not fulfilled
The pair (3,1) is not a solution of [tex]y> 2x + 1[/tex]
Now, we want to find an ordered pair that is a solution of the region, that is, that the inequality is met.
Example:
[tex](x, y) = (3,8)[/tex]
We replace:
[tex]8> 2 (3) +1\\8> 6 + 1\\8> 7[/tex]
The inequality is met.
The pair (3,8) is solution of [tex]y> 2x + 1[/tex]
Answer:
The pair (3,1) is not a solution of [tex]y> 2x + 1[/tex]
The pair (3,8) is solution of [tex]y> 2x + 1[/tex]
Question 2:
For this case, we must evaluate each of the options in the acad region:
Coordinate 1: (x, y) = (5,3)
Set 1:
[tex]y> - \frac {1} {2} x + 5\\3> - \frac {1} {2} (5) +5\\3> - \frac {5} {2} +5\\3> \frac {5} {2}[/tex]
Is fulfilled.
Set 2:
[tex]y \leq3x-2\\3 \leq3 (5) -2\\3 \leq15-2\\3 \leq13[/tex]
Is fulfilled.
Coordinate 2: (x, y) = (4,3)
Set 1:
[tex]y> - \frac {1} {2} x + 5\\3> - \frac {1} {2} (4) +5\\3> - \frac {4} {2} +5\\3> \frac {6} {2}\\3> 3[/tex]
It is not fulfilled
Set 2:
[tex]y\leq 3x-2\\3\leq3 (4) -2\\3\leq 12-2\\3\leq 10[/tex]
If it is fulfilled.
Coordinate 3: (x, y) = (3,4)
Set 1:
[tex]y> - \frac {1} {2} x + 5\\4> - \frac {1} {2} (3) +5\\4> - \frac {3} {2} +5\\3> \frac {7} {2}\\3> 3.5[/tex]
It is not true
Set 2:
[tex]y\leq 3x-2\\4\leq 3 (3) -2\\4\leq 9-2\\4\leq 7[/tex]
Is fulfilled.
Coordinate 4: (x, y) = (4,4)
Set 1:
[tex]y> - \frac {1} {2} x + 5\\4> - \frac {1} {2} (4) +5\\4> - \frac {4} {2} +5\\4> \frac {6} {2}\\4> 3[/tex]
It is not true
Region 2:
[tex]y\leq 3x-2\\4\leq 3 (4) -2\\4\leq 12-2\\4\leq 10[/tex]
Is fulfilled
Answer:
There is not a pair that is not a solution of both at the same time
