Hi, there---

YOUR PROBLEM:
Which of the following points lie in the solution set to the following system of inequalities?




A SOLUTION:
You did not provide any points to evaluate. I will show you the solution to the system of inequalities, and
you can determine if your points lie in the solution set. The solution set is all of the ordered parts that 
make both inequalities true.

I. Graph the related equation for each inequality,  and . Since the inequalities are 
both "less than or equal to," all points on these lines lie in the solution set. (See graph below.)


 

2. Now, let's find the rest of the solution set. Notice that the two graphed lines divide the coordinate plane 
into four regions. We will test one point in each region. Fortunately, if one point in the region is part of the
solution set, all the points in that region are part of the solution set. 

Conversely, if any point in a region is not part of the solution, then none of the points in the region are
part of the solution set.

The upshot is that we only need to test four points. I will call the four regions TOP, BOTTOM, LEFT, and 
RIGHT.

3. Test the TOP Region. Choose any point in the top region. I'll choose (0, 0) because the calculations are 
easy. Substitute 0 for x and 0 for y in the first inequality.

y <= x - 5
(0) <= (0) - 5
0 <= -5  FALSE

The point (0,0) is not part of the solution for the first inequality because 0 is not less than or equal to -5.
We could go on to test  (0, 0) in the second inequality. However, in order for the TOP region to be part of 
the solution set for the system of inequalities, (0, 0) must make both inequalities true. It has already failed 
the test. The TOP region is not part of the solution set. So we move on.

4. Test the BOTTOM region. Choose any point in the bottom region. I'll choose (1, -7), but you may select 
any point in the region. Substitute 1 for x and -7 for y in the first inequality.

y <= x - 5
(-7) <= (1) -5
-7 <= -4  TRUE

The point (1, -7) makes the first inequality true, so test the same point in the second inequality.

y <= -x - 4
(-7) <= -(1) - 4
-7 <= -5  TRUE

The point (1, -7) also makes the second inequality true, so the BOTTOM region is part of the solution set.

5. Use Steps 4 and 5 to test the LEFT and RIGHT region. Remember that only when your test point makes 
both inequalities true is that region part of the solution set.

I used (-3, 3) in the LEFT region, and (8, 1) in the RIGHT region. See my illustration below. 

We find that only the BOTTOM region is shaded. So, all the points in the BOTTOM region, AND all the 
points on the the lines for the equations make both inequality true, and they make up the solution set.


I'm sorry that the drawing is turned sideways, but I hope it helps. Feel free to email me if you have questions.

Mrs. F
math.in.the.vortex@gmail.com