Problem 124 - College Algebra

Question

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A city commission has proposed two tax bills. The first bill requires that a homeowner pay \(\$ 1800\) plus \(3 \%\) of the assessed home value in taxes. The second bill requires taxes of \(\$ 200\) plus \(8 \%\) of the assessed home value. What price range of home assessment would make the first bill a better deal?

Step-by-Step Solution

Answer

The first tax bill becomes a better deal when the assessed home value is more than \$32000.

Step 1: Setup The Inequalities

The first bill requires a homeowner to pay \( \$ 1800\) plus \(3\%\) of the home value, this translates into the equation: \( 1800 + 0.03x\). The second bill requires a tax of \( \$ 200\) plus \(8\%\) of the home value, this gives the equation: \( 200 + 0.08x\). To find out at what home value range the first bill becomes more advantageous, we need to setup the inequality \( 1800 + 0.03x < 200 + 0.08x\).

Step 2: Solve the Inequality

Solving the inequality involves isolating the variable. Subtract 0.03x from both sides to get 0.05x > 1600. Then divide both sides by 0.05 to solve for x. This gives the result x > 32000.

Step 3: Interpret the Answer

The result x > 32000 means that the first bill becomes a better deal when the assessed home value is more than \$32000.