What inequalities can be written for the hours and distance driven?
Practice makes perfect
We are given information about two friends driving on a road trip. Let's start by organizing the information into inequalities. We can assign variables to the hours driven by each of the friends.
Hours driven by Friend 1: y
Hours driven by Friend 2: x
We are given the average rate that each friend drives, 60 miles an hour for Friend 1 and 55 miles an hour for Friend 2. They want to drive at least 500 miles per day. We can represent this information using an inequality.
60y+55x≥500
Next, we know that they want to drive no more than 10 hours in a day. Let's write this using an inequality.
x+y ≤10
The last bit of information is that the friend that drives slower wants to drive fewer hours. We can rewrite this as an inequality.
xboundary line for the first inequality by replacing the greater than or equal sign with an equals sign and isolating the y-variable.
We can now graph the boundary line. Since the inequality is non-strict, we will use a solid line. Note that a person cannot drive for a negative number of hours, so we will only consider positive values of x and y.
Now, we can determine which region to shade using a test point. We will use the point (0,0) for simplicity. Let's substitute the values into the original inequality.
Since substituting (0,0) produced a false statement, we will shade the region that does not contain the point.
We will follow the same process to graph the other two inequalities. To graph the second inequality, x+y ≤10, we need to replace the less than or equal to sign with the equals sign and isolate the y-variable by subtracting x from both sides.
x+y=10 ⇔ y=- x+10
Since the inequality is non-strict, the line will be solid. We will use the same test point to determine which region to shade.
Since the substitution made a true statement, we will shade the region containing the test point.
Finally, we can graph the third inequality, xstrict, the line will be dashed to show that the points on the line are not part of the solution set. This time, we will use the test point (1,0) to determine where to shade, since the point (0,0) lies on the boundary line.
Since substituting (1,0) made the inequality false, we will shade the region that does not contain the test point.
The solution set is the region where all three of the shaded regions overlap.
Each point in the overlapping shaded region represents the possible hours that can be driven by each friend. For example, the point (1,8) represents the situation in which Friend 1 drives 1 hour while Friend 2 drives 8 hours.
