Question

The table below shows the cost of renting a moving van for one day from two companies. The daily charge and the charge per mile shown below:

1. Calculate the miles that you have to drive before the cost of renting a van for one day from company A is less than the cost of renting a van for one day from company B. Express in inequality.
2. Draw the graph of the inequality.
3. Find which company is less expensive of you drive $100$ miles in one day. Explain.

Short answer

a. The van from company A will have to be driven $214.28$ miles.

b. The graph is shown below:

c. Company A is less expensive if $100$ miles is driven per day.

Step-by-step solution

Step 1.  Apply the concept of Linear inequality

In this type, use the concept of making a linear inequality and use the various mathematical operation like addition, subtraction or division to get the result.

Step 2.  Finding the inequality for the situation

Since,

Cost of renting a van from company A = $\$0.35$

Daily charge of a van of company B = $\$75$.

Then, let the miles travelled by Company A be m.

So, the equation be :$0.35m<75$

Step 3.  Solving the equation to get the solution

Since, the equation is $0.35m<75$.

Simply, divide by $0.35$to both side

$\begin{array}{l}0.35m<75\\ \frac{0.35m}{0.35}<\frac{75}{0.35}\\ m<214.28\end{array}$

Hence, the van from company A will have to be driven miles before the cost of renting a van for one day from company A is less than the cost of renting a van for one day from company B.

Step 1.  Apply the concept of plotting the graph

In this type, use the concept of put the equation in the graphing calculator and find the graph.

Step 2.  Finding the inequality for the situation

Since,

The equation that have to put in the graphing calculator is $m<214.28$.

Graphical Interpretation

Following is the graph of the expression$m<214.28$ to get the proper graphical representation to know the points.

Step 1.  Apply the concept of putting the value of variable and get the answer

In this type, use the concept of putting the variable value in the equation and it gives suitable answer.

Step 2.  Finding the less expensive company

Since,

Cost of driving $100$ miles per day from company A = $100\text{miles}\times \$.35=\$35\left[\text{multiply}\right]$

Cost of driving $100$ miles per day from company B = $100\text{miles}\times \$.39=\$39\left[\text{multiply}\right]$

The cost per mile of company A is less than the cost per mile of company B.

Hence, Company A is less expensive if $100$ miles is driven per day.