a The surf shop has a weekly overhead of $2300, so to make a profit they need to earn more than $2300. Let x be the number of skimboards and y be the number of longboards. A skimboard costs $115 and a longboard $685, so we can write an expression to represent earnings for selling x skimboards and y longboards.
115x+685y dollars
Since they need to earn more than $2300 to meet their overhead, this expression must be greater than 2300. This gives us the following inequality.
115x+685y>2300
b To find how many skimboards and longboards the shop must sell each week to make a profit, let's graph the inequality from Part A. To do this, we can begin with graphing the boundary line and then we will test a point to shade the correct region.
Boundary line
To graph the boundary line, we treat the inequality as if it was an equation. Thus, we want to graph the following equation.
115x+685y=2300
Instead of rewriting it in a slope intercept form, we can find its x- and y-intercepts and connect the points with a straight line.
x-intercept
To find the x-intercept we will substitute y=0 in our equation for the boundary line.
The y-intercept of the line is approximately at y=3.36. Let's plot the points and connect them with a line! Notice that this is a boundary line of a strict inequality. It must be dashed, as it is not a part of the solution set.
Testing the point and graphing
Let's test the point (0,0) to check if it lies in the solution set!
Since 0 is not greater than 2300, the point (0,0) is not a part of the solution set. Therefore, we shade the region without this point. Also, to determine how many skimboards and longboards the shop must sell each week to make a profit, we will identify one point from the solution set.
We can tell that the point (5,3) lies in the solution set. Therefore, if the shop sells 5 skimboards and 3 longboards each week, it will make a profit.
