Free Linear Inequalities Word Problems Worksheet

While teaching high school, one of the biggest struggles I have seen students face is applying math concepts to the real-world.

This is true when it comes to solving linear inequalities word problems too. Students often find it challenging to grasp how these mathematical principles translate into practical, everyday life scenarios.

That’s why I have put together this linear inequalities word problems worksheet! My goal is to help you learn a few tips and tricks and practice applying linear inequalities to the real-world! 

What are Linear Inequality Word Problems?

Lesson plans that focus on linear inequality word problems typically show students how to apply the skills they developed while solving inequalities to the real-world. There are a wide variety of inequality applications, ranging from social studies to physical science. Regardless of the application, the idea is that you will be faced with a word problem that requires you to model the scenario using a linear inequality.

In general, linear inequality word problems describe how one quantity has to be less than or greater than another. Your goal is then to use inequality symbols and algebraic expressions to represent the scenario algebraically.

For most, the concept of a linear inequality is first introduced in middle school (although this will vary by curriculum). My daughter, for example, is in 2nd grade math and is just starting to explore these problems. Others may not see this concept until later in high school (in some cases not until the 12th grade).

How to Solve a Linear Inequality Word Problem

The best way to solve any math word problem is to start by reading the question very carefully, and linear inequality word problems are no different!

I always encourage my students to underline or highlight any key words and important information. When it comes to how to solve a linear inequality word problem, the key words usually help you understand:

• which quantities you are working with
• whether you are working with less than or greater than symbols

Once you have identified this important information, your goal is to write an algebraic expression using an inequality symbol that models the scenario. You can then solve the inequality using a similar process to what you would apply when solving one-step equations or two step equations.

There are many different ways to represent the solution to an inequality problem. Sometimes you will be asked to use a number line, which shows all the negative values or positive values that belong to a solution set. The worksheet attached below will provide you with some practice using number lines to communicate your answers to inequality word problems.

​Solving a Linear Inequality Word Problem Example

Age problems are common applications that you will see when solving linear inequality word problems. These types of problems can be simple or complex, but I wanted to start by sharing a simple one here so that you can understand the basics of how to solve linear inequality word problems.

A father is 3 times as old as his son, but three times his son’s age is less than 30. What is the oldest the son can be?

We can begin by calling out the key words that give us important information. In this case, the following two pieces of information are considered important to the problem:

• “3 times as old as”
• “less than 30”

This tells us that we will be working with a “<” symbol, and multiplication of a quantity by 3. If we let n represent the age of the son, we can set up a linear inequality as follows:

$$3n<30$$

Reading this statement in English tells us that “3 times the son’s age is less than 30”. If you head back to the original problem, that seems to match the scenario given, doesn’t it? Great! That tells us that we have a good algebraic model for our real-world problem! Let’s move on and start solving!

Remember that we can solve a linear inequality using algebra in a similar way to solving a linear equation. This means that we can add or subtract terms on both sides of the inequality symbol, and we can also multiply and divide terms on both sides of the inequality symbol.

In this case, since we are multiplying n by 3, we divide both sides by 3 to isolate n.

$$\frac{3n}{3}< \frac{30}{3}$$

$$n<10$$

This tells us that the son’s age must be less than 10 in order for the father to be three times his age but still less than 30 years old himself.

We can test this by multiplying a number greater than 10 by 3. For example, \(11 \times 3 = 33\). Notice that 33 is not less than 30. Therefore the son cannot be 11. The only values that will make this inequality statement true are values that are less than (not including) 10.

We can represent this solution on a number line by placing a hollow circle at 10 and drawing an arrow to the left toward the negative values. However, we should stop at zero since the son’s age cannot be less than zero. Note that the son must also not be equal to zero. If he were, the father would be \(3 \times 0 = 0\) as well!

If you need more practice with the algebra strategies that can be used to solve inequalities, check out this collection of solving linear inequalities worksheets.

Linear Inequalities Word Problems Worksheet

Now that you have had some practice applying your understanding of linear inequalities to solve a real-world problem, you are ready to practice! Below I have included a linear inequalities word problem worksheet that covers a variety of problem types, ranging from age problems, to a bake sale problem with pink cupcakes!

As promised, this worksheet will also provide you with practice representing the solution to a linear inequality word problem on a number line. While you should be sure to attempt every problem as independent work first, make sure that you also check the answer key! This is an important step to make sure that you fully understand each problem.

My hope is that you find this linear inequalities word problems worksheet helpful as independent work whether you are in 1st grade math, 7th grade math, or anything in between!

Download the PDF worksheet by clicking below!

Practice Solving Linear Inequalities Word Problems

I hope this linear inequalities word problems worksheet has provided you with some practice with applying this important math concept to the real-world! My goal was to share examples that cover a variety of areas of life. Hopefully solving these problems also gave you an appreciation for how linear inequalities can be used in everyday life.

Whether you first learned about linear inequality word problems in the 6th grade or are just experiencing them for the first time in the 12th grade, the most important thing you can do is practice. Solving many different types of linear inequalities word problems will help you start to recognize patterns. This will help you with the first initial step of writing the inequality as an algebraic statement, something many students find challenging! 

With consistent practice, you’ll develop a strong foundation in solving linear inequalities, enhancing your problem-solving skills and confidence in applying mathematical concepts to diverse real-world scenarios. Keep exploring and practicing, and you’ll find that handling linear inequalities becomes more intuitive over time!

If you are looking for more word problem resources, check out this linear equations word problems worksheet!

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