let

c = the money invested in city bank

s = the money invested in state bank

Bob is planning to invest up to $25000 in city bank or state bank

c + s <= 25000

he wants to invest at least 3000 dollars n city bank, but not more than 23000 dollar

3000 <= c <= 23000

since state bank does not insure more than 7000 dollars, he wants to invest no more than this amount in state ban

s <= 7000

the interest at city bank is 7% and the interest at state bank is 9% thus the total interest will be

F(c, s) = 0.07c + 0.09s

we need to find c and s so that F(c, s) has maximum

from the graph of the system of inequalities (the constrains)

c + s <= 25000

3000 <= c

c <= 23000

s <= 7000

click here to see the graph

we find the corner points of the graph: (3000,0), (3000, 7000), (18000 ,7000), (23000, 2000) and (23000, 0)

the function F(c, s) = 0.07c + 0.09s takes its maximum at a corner point. we will calculate F(c, s) at each of the corner points:

F(3000,0) = 0.07*3000 = 210

F(3000, 7000) = 0.07*3000 + 0.09*7000 = 210 + 630 = 840

F(18000, 7000) = 0.07*18000 + 0.09*7000 = 1260 + 630 = 1890

F(23000, 2000) = 0.07*23000 + 0.09*2000 = 1610 + 180 = 1790

F(23000, 2000) = 0.07*23000 = 1510

for (18000, 7000) the function F has maximum thus Bob should invest $18,000 at the city bank and $7,000 at the state bank to earn most interest.