1. **State the problem:** A car originally costs $29500. It depreciates $1680 in the first year and $475 each subsequent year. We want to find how many years it takes for the car's value to become less than half of the original price for the first time. 2. **Define variables and expressions:** - Original price: $29500$ - Half of original price: $$\frac{29500}{2} = 14750$$ - Depreciation after first year: $1680$ - Depreciation each year after first: $475$ - Let $n$ be the total number of years. 3. **Write the depreciation formula:** - After 1 year, value is $$29500 - 1680$$ - After $n$ years ($n > 1$), value is $$29500 - 1680 - 475(n-1)$$ 4. **Set inequality for value less than half:** $$29500 - 1680 - 475(n-1) < 14750$$ 5. **Solve the inequality:** $$29500 - 1680 - 475n + 475 < 14750$$ $$29500 - 1205 - 475n < 14750$$ $$28295 - 475n < 14750$$ 6. **Isolate $n$:** $$28295 - 14750 < 475n$$ $$13545 < 475n$$ 7. **Divide both sides by 475:** $$\frac{13545}{\cancel{475}} > \frac{475n}{\cancel{475}}$$ $$28.5 > n$$ 8. **Interpretation:** The value becomes less than half after more than 28.5 years, so the first integer year when this happens is $n=29$. **Final answer:** It will take 29 years for the car's price to become less than half of the original price for the first time.