We have the following equations:
[tex](1) \ y=-0.5x+5 \\ (2) \ y=-1.25x+8[/tex]
So we are asked to write a system of equations or inequalities for each region and each point.
Part a)
Region Example A
[tex]y \leq -0.5x+5 \\ y \leq -1.25x+8[/tex]
Region B.
Let's take a point that is in this region, that is:
[tex]P(0,6)[/tex]
So let's find out the signs of each inequality by substituting this point in them:
[tex]y \ (?)-0.5x+5 \\ 6 \ (?) -0.5(0)+5 \\ 6 \ (?) \ 5 \\ 6\ \textgreater \ 5 \\ \\ y \ (?) \ -1.25x+8 \\ 6 \ (?) -1.25(0)+8 \\ 6 \ (?) \ 8 \\ 6\ \textless \ 8[/tex]
So the inequalities are:
[tex](1) \ y \geq -0.5x+5 \\ (2) \ y \leq -1.25x+8[/tex]
Region C.
A point in this region is:
[tex]P(0,10)[/tex]
So let's find out the signs of each inequality by substituting this point in them:
[tex]y \ (?)-0.5x+5 \\ 10 \ (?) -0.5(0)+5 \\ 10 \ (?) \ 5 \\ 10\ \textgreater \ 5 \\ \\ y \ (?) \ -1.25x+8 \\ 10 \ (?) -1.25(0)+8 \\ 10 \ (?) \ 8 \\ 10 \ \ \textgreater \ \ 8[/tex]
So the inequalities are:
[tex](1) \ y \geq -0.5x+5 \\ (2) \ y \geq -1.25x+8[/tex]
Region D.
A point in this region is:
[tex]P(8,0)[/tex]
So let's find out the signs of each inequality by substituting this point in them:
[tex]y \ (?)-0.5x+5 \\ 0 \ (?) -0.5(8)+5 \\ 0 \ (?) \ 1 \\ 0 \ \ \textless \ \ 1 \\ \\ y \ (?) \ -1.25x+8 \\ 0 \ (?) -1.25(8)+8 \\ 0 \ (?) \ -2 \\ 0 \ \ \textgreater \ \ -2[/tex]
So the inequalities are:
[tex](1) \ y \leq -0.5x+5 \\ (2) \ y \geq -1.25x+8[/tex]
Point P:
This point is the intersection of the two lines. So let's solve the system of equations:
[tex](1) \ y=-0.5x+5 \\ (2) \ y=-1.25x+8 \\ \\ Subtracting \ these \ equations: \\ 0=0.75x-3 \\ \\ Solving \ for \ x: \\ x=4 \\ \\ Solving \ for \ y: \\ y=-0.5(4)+5=3 [/tex]
Accordingly, the point is:
[tex]\boxed{p(4,3)}[/tex]
Point q:
This point is the [tex]x-intercept[/tex] of the line:
[tex]y=-0.5x+5[/tex]
So let:
[tex]y=0 [/tex]
Then
[tex]x=\frac{5}{0.5}=10[/tex]
Therefore, the point is:
[tex]\boxed{q(10,0)}[/tex]
Part b)
The coordinate of a point within a region must satisfy the corresponding system of inequalities. For each region we have taken a point to build up our inequalities. Now we will take other points and prove that these are the correct regions.
Region Example A
The origin is part of this region, therefore let's take the point:
[tex]O(0,0)[/tex]
Substituting in the inequalities:
[tex]y \leq -0.5x+5 \\ 0 \leq -0.5(0)+5 \\ \boxed{0 \leq 5} \\ \\ y \leq -1.25x+8 \\ 0 \leq -1.25(0)+8 \\ \boxed{0 \leq 8}[/tex]
It is true.
Region B.
Let's take a point that is in this region, that is:
[tex]P(0,7)[/tex]
Substituting in the inequalities:
[tex]y \geq -0.5x+5 \\ 7 \geq -0.5(0)+5 \\ \boxed{7 \geq \ 5} \\ \\ y \leq \ -1.25x+8 \\ 7 \ \leq -1.25(0)+8 \\ \boxed{7 \leq \ 8}[/tex]
It is true
Region C.
Let's take a point that is in this region, that is:
[tex]P(0,11)[/tex]
Substituting in the inequalities:
[tex]y \geq -0.5x+5 \\ 11 \geq -0.5(0)+5 \\ \boxed{11 \geq \ 5} \\ \\ y \geq \ -1.25x+8 \\ 11 \ \geq -1.25(0)+8 \\ \boxed{11 \geq \ 8}[/tex]
It is true
Region D.
Let's take a point that is in this region, that is:
[tex]P(9,0)[/tex]
Substituting in the inequalities:
[tex]y \leq -0.5x+5 \\ 0 \leq -0.5(9)+5 \\ \boxed{0 \leq \ 0.5} \\ \\ y \geq \ -1.25x+8 \\ 0 \geq -1.25(9)+8 \\ \boxed{0 \geq \ -3.25}[/tex]
It is true
