Question

CAR SALES: For Exercise 48 and 49, use the following information.

Mrs. Lucas earns a salary of $\backslash (24,000$per year plus $1.5\%$ commission on her sales. If the average price of a car she sells is $\backslash )30,500$, about how many cars must she sell to make an annual income of at least $\$40,000$.

49. Solve the inequality and interpret the solution.

Short answer

The number of cars is$\text{x}\ge 35$.

Step-by-step solution

- Define the variable.

A variable is a term that is used to represent an unknown value.

- State the inequality.

From given, let $x$ be the number of cars she sells to make an annual income of at least $\$40,000$.

Now, the average price of a car she sells is $\$30,500$.

Total price of the cars will be $\$30,500\times x$.

She has $1.5\%$of commission on her sells.

That is,

That is equal to $\frac{1.5}{100}\times \$30,500\times x$.

She also has a salary of $\$24,000$

The inequality is:

$\$24,000+\left(\frac{1.5}{100}\times \$30,500\times x\right)\ge \$40,000$

- Solve the inequality.

Simplify the inequality as shown below:

$\begin{array}{c}\$24,000+\left(\frac{1.5}{100}\times \$30,500\times x\right)\ge \$40,000\\ \$24,000+457.5x-\$24,000\ge \$40,000-\$24,000\\ 457.5x\ge \$16,000\\ x\ge 34.9\approx 35\end{array}$

Therefore, the number of cars needed to be sold is $x\ge 35$.

- State the interpretation.

As $x>35$. Thus, Mrs. Lucas needs to sell minimum 35 cars so that she can have an annual income of at least $\$40,000$.