Examples of Inequalities Word Problems With Solutions

Inequalities are a powerful tool that is taught in high school mathematics that allows us to solve a variety of real-life situations where a range of multiple solutions are possible.

Inequalities help us model scenarios with constraints such as budgeting, scheduling, or meeting minimum requirements, while guiding us toward finding the most practical range of solutions.

My goal is to walk you through a few key tips and examples to master solving linear inequality word problems with solutions that show you each step!

What Are Inequality Word Problems?

An inequality is a mathematical statement that shows the relationship between two expressions using symbols.

Inequality statements involve one of the following symbols:

< (less than)
> (greater than)
≤ (less than or equal to)
≥ (greater than or equal to)

Unlike equations, inequalities provide a range of possible values that satisfy specific conditions. Inequalities word problems require us to find the set of solutions that make an inequality statement true.

In real world problems, linear inequalities are commonly used to represent scenarios with limits, such as “at least 5” or “no more than 10.”

We can translate these statements into mathematical models by identifying key words and assigning appropriate variable names to unknowns.

My Top Tips for Solving Inequality Word Problems

When solving an inequality algebraically, we apply the same algebraic equation solving strategies that you have learned!

Before diving into some examples of inequalities word problems, I want to share some strategies for solving these problems effectively. These tips are similar to my strategies for solving math word problems.

Choose Clear Variable Names: Use names that represent the context of the problem. For example, we might choose to let $x$ represent the quantity given in the problem. Depending on the scenario, this could represent the number of items, an average speed, etc.
Identify the Key Words Used: Be on the lookout for phrases like “at least,” “no more than,” and “maximum”. These phrases give you specific information about how to set up your inequality. I always recommend underlining these in the problem so that you can clearly see all of the key words while formulating a plan.
Translate the Word Problem to a Mathematical Model: Use the appropriate symbol that matches the key words you underlined. Write an inequality that captures the relationships in the problem.
Check Your Work: After solving the inequality word problem, substitute a few test values into the original problem to ensure they satisfy all conditions.

Let’s take a look at how these tips for solving inequality word problems can be used to solve some problems!

Examples of Inequalities Word Problems

Let’s dig into some real-life situation examples of inequalities word problems. I have also provided a solution answer key for each of the problems with detailed explanations of each step!

Example 1: Budgeting for a Scavenger Hunt

A group is organizing a scavenger hunt and must spend at least $50 on prizes to qualify for a group discount. Each prize costs $8. How many prizes do they need to buy to meet this minimum spending requirement?

Solution:

We can begin by letting $x$ represent the number of prizes. We can write a statement that uses the greater than or equal to symbol since we know that the group needs to spend a minimum of $50 (or at least $50).

Since each prize costs $8, we know that we will be multiplying 8 by our variable of $x$. This results in the following inequality statement:

\[8x \geq 50\]

We can solve by dividing both sides of the inequality by 8:

\[\begin{split} x &\geq \frac{50}{8} \\ \\ x & \geq 6.25 \end{split}\]

Since we know that we can’t have fractional prizes, we must round up to the next whole number to ensure the spending meets the minimum requirement.

Therefore, the group must buy at least 7 prizes.

Example 2: Average Speed Problem

A student drives to school and wants to arrive on time for an 8:00 AM class. If the school is 20 miles away and they leave at 7:30 AM, what minimum speed must they maintain to avoid being late?

Solution:

Let’s start by letting $s$ represent the speed in miles per hour.

We know that there is a relationship between distance, average speed, and time. This relationship is $time=\frac{distance}{speed}$. We can use this relationship to create an inequality statement.

Since we are told that the student must arrive within a half hour, we know that we will be using a less than or equal to symbol in our statement. Substituting our distance and time into our equation results in:

\[\begin{split} \frac{distance}{speed} &\leq 0.5 \\ \\ \frac{20}{s} &\leq 0.5 \\ \\ \frac{20}{0.5} &\leq s \\ \\ 40 &\leq s \end{split}\]

We can rewrite this solution as $s \geq 40$ to show that speed must be greater than or equal to 40 miles per hour.

Therefore, we can conclude that the student must drive at least 40 miles per hour to arrive on time!

Example 3: Determining the Number of Girls in a Club

In a school club, there are at least twice as many boys as girls. If there are 12 boys, determine the number of girls that there could be in the club.

Solution:

Let’s begin by letting the variable $g$ represent the number of girls. We know that the number of boys is twice the number of girls. We represent this as $2g$ in our inequality statement. In total, this amount has to be less than or equal to 12.

We set up our inequality statement as:

\[\begin{split} 2g &\leq 12 \\ \\ g &\leq \frac{12}{2} \\ \\ g &\leq 6 \end{split}\]

Therefore, we can conclude that there can be up to 6 girls in the club.

Looking for some more practice? Check out my collection of linear inequalities worksheets! Then, use this inequalities word problem worksheet to solidify your understanding!

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