SOLUTION: My daughter has been struggling with this algebra problem and I couldn't help help her. Please help! "Carlo needs to buy at least eight flowers and has $15 to spend. Roses cost

For the graph:

Let x = no. of roses
Let y = no. of carnations 

 x + y > 8   (Guarantees there are at least 8 flowers)
3x + y < 15  (Guarantees that the total cost does not exceed $15)
     x > 0   (Guarantees there cannot be a negative number of roses)
     y > 0   (Guarantees there cannot be a negative number of carnations)


Form the equations of the boundary lines by replacing the 
inequality symbols by = signs.

These equations are:

 x + y = 8    This line has intercepts (0,8) and (8,0)
3x + y = 15   This line has intercepts (0,15) and (5,0)
     x = 0
     y = 0

The last two equations are just the equations of the
y and x axes respectively.  They just mean that 
everything must be in the first (upper right) quadrant,
i.e., no negative velues for x or y

Graph the first line using the intercepts:



All points (x,y) ON or ABOVE that line represent
combinations of x roses and y combinations which
consist of 8 or mor roses. However many of those
points, in fact most of them, represent combinations
that would cost more than $15.

Now if we draw only the second line, using the intercepts:  

 

Now all points (x,y) ON or BELOW that line represent
combinations of flowers whose costs do not exceed
$15.

So now we put both lines on the same graph:

 

Only those points (x,y) in the triangle marked XXX are
the possible combinations that he may buy with no more 
than $15 yet have at least 8 flowers.

Or, erasing the unused portions, we have only this 
triangle (it is called the "feasible region")



The only problem here is that we can't have fractions of flowers, 
and most of the points in that region involve fractions, so we
must grid off the feasible region like this:



and the only possibilities are the points which are corners
of the squares, i.e., both of whose coordinates are integers.
These are called "lattice points". You can count the corners
of squares included in or on the triangle. There are 20 of them: 

1. (0,8) He could buy 0 roses, 8 carnations, 8 flowers for $8.
2. (0,9) He could buy 0 roses, 9 carnations, 9 flowers for $9.
3. (0,10) He could buy 0 roses, 10 carnations, 10 flowers for $10.
4. (0,11) He could buy 0 roses, 11 carnations, 11 flowers for $11.
5. (0,12) He could buy 0 roses, 12 carnations, 12 flowers for $12.
6. (0,13) He could buy 0 roses, 13 carnations, 13 flowers for $13.
7. (0,14) He could buy 0 roses, 14 carnations, 14 flowers for $14.
8. (0,15) He could buy 0 roses, 15 carnations, 15 flowers for $15.
9. (1,7) He could buy 1 rose and 7 carnations, 8 flowers for $10.
10. (1,8) He could buy 1 rose and 8 carnations, 9 flowers for $11.
11. (1,9) He could buy 1 rose and 9 carnations, 10 flowers for $12.
12. (1,10) He could buy 1 rose and 10 carnations, 11 flowers for $13.
13. (1,11) He could buy 1 rose and 11 carnations, 12 flowers for $14.
14. (1,12) He could buy 1 rose and 12 carnations, 13 flowers for $15.
15. (2,6) He could buy 2 roses and 6 carnations, 8 flowers for $12.
16. (2,7) He could buy 2 roses and 7 carnations, 9 flowers for $13.
17. (2,8) He could buy 2 roses and 8 carnations, 10 flowers for $14.
18. (2,9) He could buy 2 roses and 9 carnations, 11 flowers for $15.
19. (3,5) He could buy 3 roses and 5 carnations, 8 flowers for $14.
20. (3,6) He could buy 3 roses and 6 carnations, 9 flowers for $15.

Edwin