Higher Order Thinking Andrea went to the store to buy a sweater that was on sale for \( 40 \% \) off the original price. It was then put on clearance at an additional \( 25 \% \) off the sale price. She also used a coupon that saved her an additional \( \$ 5 \). Andrea did not spend more than \( \$ 7.60 \) for the sweater. What are the possible values for the original price of the sweater?

To solve this, let's denote the original price of the sweater as \( P \). First, we calculate the price after the 40% discount, which is \( P - 0.4P = 0.6P \). Then, we apply the additional 25% discount on the sale price: \[ 0.6P - 0.25(0.6P) = 0.6P - 0.15P = 0.45P. \] After applying the coupon, the final price Andrea pays is: \[ 0.45P - 5. \] According to the information, this amount cannot exceed $7.60: \[ 0.45P - 5 \leq 7.60. \] Adding 5 to both sides gives us: \[ 0.45P \leq 12.60. \] Now, dividing both sides by 0.45 leads to: \[ P \leq \frac{12.60}{0.45} \approx 28. \] This means that the original price of the sweater can be anything from 0 (it can't be negative) up to $28. Thus, the possible values for the original price of the sweater should be \( 0 < P \leq 28 \). But watch out! If you happen to think about it as a real-world scenario, it's important to keep an eye on the psychological pricing—nobody wants to feel they overpaid, even if it was on sale!