Systems of Linear Inequalities, Word Problems - Examples

Word Problems with Systems of Linear Inequalities

Solving word problems with systems of linear inequalities involves 2 steps:

1. Translate the word problem into mathematical expressions. We're finding a system of linear inequalities.

2. Solve! One strategy you can use to find solutions is to graph each inequality. Then, use the overlapping areas of the graphs to identify the solutions to the word problem.

Let's look at an example:

STEP 1: Translate the word problem into inequalities.

We're going to use $x$ for the number of pizzas and $y$ for the number of pounds of cookies.

Let's recap:

$$\begin{align} 9x + 5y &\leq 50 \\ x &\geq 3 \\ y &\geq 2 \end{align}$$

The first inequality tells us that, all together, Pilar can spend up to (and including) $\$50$.

The second inequality tells us that Pilar needs $3$ or more pizzas.

The third inequality tells us that Pilar needs $2$ or more pounds of cookies.

STEP 2: Solve!

We are going to use graphing to solve these inequalities, but you could use another strategy instead!

Recall that inequalities using $<$ and $>$ should have dotted lines, while inequalities using $\leq$ and $\geq$ should have solid lines. Also, we will be shading our graph to show the solutions to each inequality. (Look back over solving systems by graphing if you need a refresher!)

We are looking for the area where all three shaded areas overlap. These are the values that make all three inequalities in the system true.

The solution is the little triangle in the upper right quadrant. This means that any coordinate values of $x$ and $y$ in that area are solutions to our inequalities. In other words, Pilar can buy any combination of pizza and cookies represented in that triangle.

For instance, the point $(4, 3)$ is included in this little triangle. Since we used $x$ to represent pizzas and $y$ to represent cookies, this means that Pilar can buy $4$ pizzas and $3$ pounds of cookies while staying in budget.

(Video) Solving Inequalities: Word Problems

by Anusha Rahman

This video by Anusha Rahman walks you through an example word problem of how to solve a system of inequalities.

Summary

Example: Jackson is having a party for his friends on Saturday. Jackson expects at least 7 people to show up on Saturday. He is trying to figure out the total costs of all of the pizzas. He calls his local pizza place, Meredith’s pizza, who says that the cost of his pizzas will be less than 5 more than 2 times the number of people he needs pizza for.

Step 1: Establish variables $$x=\text{the number of people}$$ $$y=\text{the cost of all pizzas}$$

Step 2: Set up the inequalities $$x \ge 7$$ $$y < 5 +2x$$

Step 3: Graph the inequalities to find the solution

Graph of: $x \ge 7$

Graph of $y=5+2x$

This is the graph of the line, $y=5+2x$. In order to graph the inequality, $y < 5 +2x$, we have to figure out what area on the graph to shade. To do that, we test out some points and see whether the inequality remains true!

Let's test the point $(10,2)$. $$2<2(10)+5$$ $$2<20 + 5$$ $$2<25$$ This statement is true! So we will probably shade in this area, the bottom area of the graph. But, let's just check one more point to make sure.

Let's test the point $(-10,2)$. $$2<2(-10)+5$$ $$2<-20+5$$ $$2<-15$$ This statement is false.

So, we can see in the above graph that the area under the dotted line is shaded. This is the graph of $y < 5 +2x$.

Now, let's put both inequalities on the same graph to see the area of intersection!

The area shaded in purple is the solution to the system of inequalities! Because Jackson probably won't have negative friends or pay negative dollars, we can assume $x > 0$ and $y>0$.

Jackson won’t know for certain what the price of his pizzas will be. Why? Because Jackson doesn’t know how many of his friends will be coming to his party. Especially when solving real-world problems, a system of inequalities is really helpful when we’re looking at something in the future, and we don’t have exact numbers, but we have a general idea, like Jackson does.

Systems of Linear Inequalities: Word Problems

When you are presented with a word problem that deals with inequalities, it may seem daunting at first to solve the problem. The good news is that there are two simple steps to follow in order to solve systems of linear inequalities!

Let's walk through an example problem together.

Example

Jenn and John want to go on a road trip together. Jenn drives an average of $50$mph, and John drives an average of $45$mph. They want to drive less than $20$ hours, but at least $500$ miles. Write a system of linear inequalities to figure out how many miles John and Jenn must drive to make that happen.

Step 1: Translate the word problem into inequalities.

Let's start with establishing our variables. $$x=\text{Jenn}$$ $$y=\text{John}$$

We know that Jenn and John want to drive at least $500$ miles. At least generally means greater than or equal to. So if we incorporate the speed at which the two of them drive, we can say that:

$$50x+45y \geq 500$$

We also know that, combined, John and Jenn want to drive less than $20$ hours. So, we can say that: $$x+y=20$$

Therefore, our system of linear equations is:

$$50x+45y \geq 500$$ $$x+y=20$$

Step 2: Solve the inequalities.

Let's solve these inequalities by graphing. Here is a graph of the two inequalities.

$50x+45y \geq 500$ is represented by the red shading. $x+y=20$ is represented by the blue shading. The overlap, or the solution, is represented by the purple shading.