Exercise 36 Page 261

For this exercise we can write and graph a system of inequalities. The system will have three inequalities.

• Inequality (I) represents the total amount of time driving by us and our friend.
• Inequality (II) represents the distance covered given speed and time.
• Inequality (III) represents the fact that our friend drives more hours than us.

For all three inequalities we will let $x$ represent the number of hours we drive and $y$ represent the number of hours our friend drives.

Writing the Inequalities

Inequality (I): It is given that we plan to drive less than 15 hours. We can write an inequality to represent this situation in the following way. Inequality (II): It is given that we drive $70$ mph and our friend drives $60$ mph. The phrase at least can be interpreted as greater than or equal to and represented with $\ge .$ Then, we have the following inequality. Inequality (III): It is given that our friend drives more than us. We can write the following equality. With these inequalities we can write the following system.

Solving the System

To determine the number of hours we friend can drive we can graph the system created above. The solution to the system will be the intersection of the solution sets of all three inequalities. Let's graph each inequality on its own first.

Graphing Inequality (I)

To graph the first inequality we need to first graph the boundary line. We need to write the inequality as though it were a line in slope-intercept form. Note, because the symbol is $<$ our boundary line should be dashed. Now we need to decide which side of the line should be shaded. We can substitute any arbitrary point into the inequality to check if it is a solution. For simplicity, let's use $(0,0).$ Because $(0,0)$ is a solution to the inequality, we should shade the side of the line containing the point. Also, we should restrict the domain and range to reflect the fact that we cannot drive a negative amount of hours.

Graphing Inequality (II)

To graph the second inequality we will follow a similar process as we did with the first inequality. First, we need to graph the boundary line. Note, because the symbol is $\ge$ our boundary line should be solid. Let's start by rewriting the inequality so that $y$ is isolated. Note, because the symbol is $\ge$ our boundary line should be solid. As before, we test which side to shade by substituting $(0,0)$ into the inequality. Because $(0,0)$ is not a solution to the inequality we should shade the side of the line that does not contain the point. Again, we restrict the domain and range to reflect the fact that we cannot drive a negative amount of hours.

Graphing Inequality (III)

To graph the third and final inequality we will once again begin by graphing the boundary line. Note, because the symbol is $>$ our boundary line should be dashed.

Notice that $y=x$ goes through the origin so we cannot use this point to test which side to shade. Let's instead use $(6,14)$ as a test point. Because $(6,14)$ is a solution to the inequality, we should shade the side of the line containing the point. Again, we restrict the domain and range as before.

Combining the Solution Sets

If we show all of these individual solution sets on the same coordinate plane, we have the following graph.

Any point contained in the above shaded area will be a solution for how many hours we and our friend can drive in a day without driving more than $15$ hours and going at least $600$ miles. Let's write some example solutions.