A shopper plans to purchase an item that has a listed price greater than $120 and can use any one of the three coupons. Coupon A gives $20\%$ off the listed price, Coupon B gives $40 off the listed price, and Coupon C gives $30\%$ off the amount by which the listed price exceeds $120. Let $x$ and $y$ be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or Coupon C. What is $y-x$?