We've updated our
Privacy Policy effective December 15. Please read our updated Privacy Policy and tap

Study Guides > Intermediate Algebra

Read: Applications of Systems of Linear Inequalities

Learning Objectives

  • Write and graph a system that models the quantity that must be sold to achieve a given amount of sales
  • Write a system of inequalities that represents the profit region for a business
  • Interpret the solutions to a system of cost/ revenue inequalities
In our first example we will show how to write and graph a system of linear inequalities that models the amount of sales needed to obtain a specific amount of money.

Example

Cathy is selling ice cream cones at a school fundraiser. She is selling two sizes: small (which has 11 scoop) and large (which has 22 scoops). She knows that she can get a maximum of 7070 scoops of ice cream out of her supply. She charges $3 for a small cone and $5 for a large cone. Cathy wants to earn at least $120 to give back to the school. Write and graph a system of inequalities that models this situation.

Answer: First, identify the variables. There are two variables: the number of small cones and the number of large cones.

s = small cone

l = large cone

Write the first equation: the maximum number of scoops she can give out. The scoops she has available (70)(70) must be greater than or equal to the number of scoops for the small cones (s) and the large cones (2(2l) she sells.

s+2l70s+2l\le70

Write the second equation: the amount of money she raises. She wants the total amount of money earned from small cones (3<i>s</i>)(3<i>s</i>) and large cones (5<i>l</i>)(5<i>l</i>) to be at least $120.

3s+5l1203s+5l\ge120

Write the system.

{s+2l703s+5l120s>=0l>=0\begin{cases}s+2l\le70\\3s+5l\ge120\\s>=0\\l>=0\end{cases}

Now graph the system. The variables x and y have been replaced by s and l; graph s along the x-axis, and l along the y-axis. First graph the region  <i>s</i>+2<i>l</i>70<i>s</i> + 2<i>l</i> ≤ 70. Graph the boundary line and then test individual points to see which region to shade. We only shade the regions that also satisfy x>=0,y>=0x>=0, y>=0. The graph is shown below. image027 Now graph the region 3s+5l1203s+5l\ge120 Graph the boundary line and then test individual points to see which region to shade. The graph is shown below. image028 Graphing the regions together, you find the following: image029 And represented just as the overlapping region, you have:image030

Answer

The region in purple is the solution. As long as the combination of small cones and large cones that Cathy sells can be mapped in the purple region, she will have earned at least $120 and not used more than 7070 scoops of ice cream.

In a previous example for finding a solution to a system of linear equations, we introduced a manufacturer’s cost and revenue equations: Cost: y=0.85x+35,000y=0.85x+35,000 Revenue: y=1.55xy=1.55x x0,y0x\ge0,y\ge0 The cost equation is shown in blue in the graph below, and the revenue equation is graphed in orange.The point at which the two lines intersect is called the break-even point, we learned that this is the solution to the system of linear equations that in this case comprise the cost and revenue equations. Note how the lines shown only represent where x0,y0x\ge0, y\ge0. It's easy to forget to include this part in the graph. The shaded region to the right of the break-even point represents quantities for which the company makes a profit. The region to the left represents quantities for which the company suffers a loss. In the next example, you will see how the information you learned about systems of linear inequalities can be applied to answering questions about cost and revenue. A graph showing money in dollars on the y axis and quantity on the x axis. A line representing cost and a line representing revenue cross at the break-even point of fifty thousand, seventy-seven thousand five hundred. The cost line's equation is C(x)=0.85x+35,000. The revenue line's equation is R(x)=1.55x. The shaded space between the two lines to the right of the break-even point is labeled profit. Note how the blue shaded region between the Cost and Revenue equations is labeled Profit. This is the "sweet spot" that the company wants to achieve where they produce enough bike frames at a minimal enough cost to make money. They don't want more money going out than coming in!

Example

Define the profit region for the skateboard manufacturing business using inequalities, given the system of linear equations: Cost: y=0.85x+35,000y=0.85x+35,000 Revenue: y=1.55xy=1.55x x0,y0x\ge0, y\ge0

Answer: We know that graphically,  solutions to linear inequalities are entire regions, and we learned how to graph systems of linear inequalities earlier in this module. Based on the graph below and the equations that define cost and revenue, we can use inequalities to define the region for which the skateboard manufacturer will make a profit.  Again, not how only the region for x0,y0x\ge0, y\ge0 is included. Cost/ Revenue with Profit Let's start with the revenue equation.  We know that the break even point is at (50,000,77,500)(50,000, 77,500) and the profit region is the blue area.  If we choose a point in the region and test it like we did for finding solution regions to inequalities, we will know which kind of inequality sign to use. Let's test the point (65,00,100,000)\left(65,00,100,000\right) in both equations to determine which inequality sign to use. Cost:

y=0.85x+35,000100,000 ? 0.85(65,000)+35,000100,000 ? 90,250\begin{array}{l}y=0.85x+{35,000}\\{100,000}\text{ ? }0.85\left(65,000\right)+35,000\\100,000\text{ ? }90,250\end{array}

We need to use > because 100,000100,000 is greater than 90,25090,250 The cost inequality that will ensure the company makes profit - not just break even - is y>0.85x+35,000y>0.85x+35,000 Now test the point in the revenue equation: Revenue:

y=1.55x100,000 ? 1.55(65,000)100,000 ? 100,750\begin{array}{l}y=1.55x\\100,000\text{ ? }1.55\left(65,000\right)\\100,000\text{ ? }100,750\end{array}

We need to use < because 100,000100,000 is less than 100,750100,750 The revenue inequality that will ensure the company makes profit - not just break even - is y<1.55xy<1.55x The systems of inequalities that defines the profit region for the bike manufacturer:

y>0.85x+35,000y<1.55x\begin{array}{l}y>0.85x+35,000\\y<1.55x\end{array}

Answer

The cost to produce 50,00050,000 units is $77,500, and the revenue from the sales of 50,00050,000 units is also $77,500. To make a profit, the business must produce and sell more than 50,00050,000 units. The system of linear inequalities that represents the number of units that the company must produce in order to earn a profit is: y>0.85x+35,000y<1.55x\begin{array}{l}y>0.85x+35,000\\y<1.55x\end{array}

In the following video you will see an example of how to find the break even point for a small sno-cone business. https://youtu.be/qey3FmE8saQ And here is one more video example of solving an application using a system of linear inequalities. https://youtu.be/gbHl6K-dJ8o We have seen that systems of linear equations and inequalities can help to define market behaviors that are very helpful to businesses.  The intersection of cost and revenue equations gives the break even point, and also helps define the region for which a company will make a profit.

Licenses & Attributions