This week we are learning about two-variable inequalities as they pertain to algebraic expressions. The inequality can be graphed to show the values included in and excluded from a given range of numbers. Solving for inequalities such as these is a critical skill in many trades which can save or cost a company a lot of time and money. Ozark Furniture Company can obtain at most 3000 board feet of maple lumber for making its classic and modern maple rocking chairs. A classic maple rocker requires 15 board feet of maple, and a modern rocker requires 12 board feet of maple. Write an inequality that limits the possible number of maple rockers of each type that can be made, and graph the inequality in the first quadrant. First I must assign a variable to each type of rocker Ozark Furniture makes. Let c = the number of classic rockers
Let m = the number of modern rockers
It takes 15 board feet of lumber for each classic rocker so I will use 15c in my equation. Likewise, I will use 12m for the 12 board feet of lumber in the modern rocker. The maximum amount of lumber Ozark can obtain is 3000 board feet. Therefore, my equation will look like this: 15c + 12m ≤ 3000
If I call c the independent variable (on the horizontal axis) and m the dependent variable (graphed on the vertical axis) then I can graph the equation using the intercepts. The c-intercept is determined when m = 0:
15c ≤ 3000
c ≤ 200
The c-intercept is (200,0).
The m-intercept is found when c = 0:
12m ≤ 3000
m ≤ 250
The m-intercept is (0, 250).
Since this inequality is “less than or equal to”, the graphed line will be solid, sloping downward from left to right within the first quadrant of the graph. The shaded section will cover the area from the line towards the origin, stopping at the respective axes.
Consider the point (50,100) on my graph. It is well inside the shaded area meaning that Ozark Furniture could easily fill this order for 50 classic and 100 modern rocking chairs. 50(15) + 100(12) = 1920 board feet of lumber leaving 1080 board feet remaining. Now consider the point (150,100) on the graph. It is outside the shaded area which indicates that this order of 150 classic rockers and 100 modern rockers would require more lumber than the company can acquire. 150(15) + 100(12) = 3450 board feet of lumber which is 450 board feet more than they have. Ozark cannot fulfill this order.
Consider the point (100, 125) now. This point falls directly on the line on the graph indicating that the company would have just enough lumber to fill this order with none to spare. 100(15) + 125(12) = 3000 board feet of maple.
If Ozark receives a faxed order for 175 modern rockers and 125 classic rockers, they would not be able to fulfill the order. When we plot the order on the graph (125, 175) we can clearly see it is well outside the shaded area. 125(15) + 175(12) = 3975 board feet of lumber required.
They would need an additional 975 board feet of lumber to complete the order.
These scenarios clearly show how important it is to be able to solve an inequality with two variables.