The value \( V \) (in dollars) of a vehicle depends on the miles \( x \) that it has been driven. This is given in the formula \( V=22,000-0.25 x \). After one year, the value of a vehicle is between \( \$ 12,000 \) and \( \$ 15,000 \). Which range of miles driven corresponds to this range of values based on the given formula?

To determine the range of miles driven (\( x \)) corresponding to the vehicle's value (\( V \)) being between \$12,000 and \$15,000, we'll use the given formula: \[ V = 22,000 - 0.25x \] **Step 1: Set Up the Inequalities** We want \( V \) to satisfy: \[ 12,000 \leq 22,000 - 0.25x \leq 15,000 \] **Step 2: Solve the Lower Bound** \[ 12,000 \leq 22,000 - 0.25x \] Subtract 22,000 from both sides: \[ -10,000 \leq -0.25x \] Multiply both sides by \(-1\) (remember to reverse the inequality sign): \[ 10,000 \geq 0.25x \] Multiply both sides by 4: \[ 40,000 \geq x \quad \text{or} \quad x \leq 40,000 \] **Step 3: Solve the Upper Bound** \[ 22,000 - 0.25x \leq 15,000 \] Subtract 22,000 from both sides: \[ -0.25x \leq -7,000 \] Multiply both sides by \(-1\) (again, reversing the inequality sign): \[ 0.25x \geq 7,000 \] Multiply both sides by 4: \[ x \geq 28,000 \] **Step 4: Combine the Results** From the above, we have: \[ 28,000 \leq x \leq 40,000 \] **Conclusion** The range of miles driven corresponding to the vehicle's value being between \$12,000 and \$15,000 is **28,000 to 40,000 miles**. **Answer:** After simplifying, the miles driven must be at least 28 000 and at most 40 000 miles.