1 nickle = 5 cents

1 dime = 10 cents

1 quarter = 25 cents

let

n = the number of nickels, n>0, n is integer

d = the number of dimes, d>0, d is integer

q = the number of quarters, q>0, q is integer

Betty has $1.50 = 150 cents

5n + 10d + 25q = 150 divide both sides by 5

n + 2d + 5q = 30

she has 2 less nickles than she has dimes

n = d - 2

and 2 more quarters then she has dimes

q = d + 2

by solving the system of equations

n + 2d + 5q = 30

n = d - 2

q = d + 2

we find

n = 2.75

d = 0.75

q = 4.75

click here to see the step by step solution of the system of equations

since n, d and q need to be integers we conclude that the problem has no solutions.