Given below are the graphs of two lines, y=-0.5 + 5 and y=-1.25x + 8 and several regions and points are shown. note that c is the region that appears completely white in the graph. can someone me with both a and b asap

We have the following equations:



So we are asked to write a system of equations or inequalities for each region and each point.

Part a)

Region Example A



Region B.

Let's take a point that is in this region, that is:



So let's find out the signs of each inequality by substituting this point in them:



So the inequalities are:



Region C.

A point in this region is:



So let's find out the signs of each inequality by substituting this point in them:



So the inequalities are:



Region D.

A point in this region is:



So let's find out the signs of each inequality by substituting this point in them:



So the inequalities are:



Point P:

This point is the intersection of the two lines. So let's solve the system of equations:



Accordingly, the point is:



Point q:

This point is the of the line:



So let:



Then



Therefore, the point is:



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:



Substituting in the inequalities:



It is true.

Region B.

Let's take a point that is in this region, that is:



Substituting in the inequalities:



It is true

Region C.

Let's take a point that is in this region, that is:



Substituting in the inequalities:



It is true

Region D.

Let's take a point that is in this region, that is:



Substituting in the inequalities:



It is true