Example

Cathy is selling ice cream cones at a school fundraiser. She is selling two sizes: small (which has [latex]1[/latex] scoop) and large (which has [latex]2[/latex] scoops). She knows that she can get a maximum of [latex]70[/latex] scoops of ice cream out of her supply. She charges [latex]$3[/latex] for a small cone and [latex]$5[/latex] for a large cone. Cathy wants to earn at least [latex]$120[/latex] 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 [latex](70)[/latex] must be greater than or equal to the number of scoops for the small cones (s) and the large cones [latex](2[/latex]l) she sells.

[latex]s+2l\le70[/latex]

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

[latex]3s+5l\ge120[/latex]

Write the system.

[latex]\begin{cases}s+2l\le70\\3s+5l\ge120\\s>=0\\l>=0\end{cases}[/latex]

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 The variables x and y have been replaced by s and l; graph s along the x-axis, and l along the y-axis.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

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 [latex]$120[/latex] and not used more than [latex]70[/latex] scoops of ice cream.

Example

Define the profit region for the skateboard manufacturing business using inequalities, given the system of linear equations: Cost: [latex]y=0.85x+35,000[/latex] Revenue: [latex]y=1.55x[/latex] [latex-display]x\ge0, y\ge0[/latex-display]

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 [latex]x\ge0, y\ge0[/latex] is included. Let's start with the revenue equation.  We know that the break even point is at [latex](50,000, 77,500)[/latex] 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 [latex]\left(65,00,100,000\right)[/latex] in both equations to determine which inequality sign to use. Cost:

[latex]\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}[/latex]

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

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

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

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

Answer

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