Explanation:
3/4 is the original x pounds of fudge contain nuts.
in short
3x/4 pounds contain nuts.
now, additional 12 3/4 pounds of fudge with nuts are made, bringing the total of pounds of fudge with nuts to at least 29 5/8.
(a)
so, the inequality is
3x/4 + 12 3/4 >= 29 5/8
the reference is x : the original number of pounds of fudge (with or without nuts) Monday morning.
(b)
to solve we need to convert all numbers to real fractions and bring all of them to the save denominator (8).
3x/4 = 2×3x/(2×4) = 6x/8
12 3/4 = 12 6/8 = (12×8 + 6)/8 = 102/8
29 5/8 = (29×8 + 5)/8 = 237/8
6x/8 + 102/8 >= 237/8
now we can get rid of the fractions by multiplying everything by 8
6x + 102 >= 237
6x >= 135
x >= 135/6 = 22 3/6 = 22 1/2 pounds
there were at least 22 1/2 pounds of fudge in the shop Monday morning.
(c)
since there are only 1-pound boxes (and no part-pound boxes), and we assume we cannot mix the types of fudge, we need to find first how many pounds there were of each type :
22 1/2 pounds in total
3x/4 pounds with nuts
that leaves x - 3x/4 = 4x/4 - 3x/4 = x/4 pounds for fudge without nuts.
22 1/2 = (22×2 + 1)/2 = 45/2 pounds = x
3(45/2)/4 = 3×45/(2×4) = 135/8 = 16 7/8 pounds of fudge with nuts.
45/2 / 4 = 45/(2×4) = 45/8 = 5 5/8 pounds of fudge without nuts.
so, there were at least 16 boxes of fudge with nuts and 5 boxes with fudge without nuts.
in total at least 16 + 5 = 21 boxes of fudge Monday morning.