Solving word problems with linear equations and inequalities is one of the most practical skills you’ll use in algebra. It helps you turn real-life situations—like budgeting, comparing rates, and calculating time—into math problems you can solve.
In this article, you’ll learn how to set up and solve word problems using linear equations and inequalities. This connects to what you’ve already learned about expressions, equations, graphing lines, and shading inequality regions.
Why Word Problems Matter
You may wonder why algebra teachers are always talking about word problems. The truth is: this is where math meets the real world. Whether you’re planning a trip, saving money, or managing time, you’re using math to make decisions.
Word problems help you:
- Translate situations into mathematical language
- Think critically about quantities and relationships
- Choose the right kind of equation or inequality to model the problem
Strategy for Solving Word Problems
Here’s a step-by-step strategy to follow:
- Read carefully and understand the situation
- Define the variable(s) — what are you solving for?
- Write an equation or inequality to represent the relationships
- Solve the equation or inequality
- Answer the question in a complete sentence
- Check your work and make sure your answer makes sense
Example 1: Linear Equation Word Problem
Problem:
Jamie earns $12 per hour at her part-time job. She wants to save $300 for a school trip. How many hours does she need to work?
Step 1: Let h = number of hours Jamie needs to work
Step 2: Write the equation: 12h = 300
Step 3: Solve the equation: h = 25
Answer: Jamie needs to work 25 hours to save $300.
Example 2: Linear Inequality Word Problem
Problem:
A school bus can hold up to 72 students. If there are already 50 students on board, how many more can get on without going over the limit?
Step 1: Let x = number of additional students
Step 2: Write the inequality: 50 + x ≤ 72
Step 3: Solve: x ≤ 22
Answer: Up to 22 more students can get on the bus.
Example 3: Writing Inequalities from Situations
Problem:
You are saving money to buy a game console that costs $400. You already have $120 and plan to save $25 per week. Write an inequality to represent how many weeks you need to save.
Step 1: Let w = number of weeks
Step 2: Write the inequality: 120 + 25w ≥ 400
This inequality means your savings must be greater than or equal to $400.
Example 4: Comparing Rates with Equations
Problem:
Tasha is renting a bike for a day. One shop charges a flat fee of $10 plus $5 per hour. Another shop charges $7 per hour with no flat fee. After how many hours will both shops cost the same?
Step 1: Let h = number of hours
Step 2: Write equations:
Shop A: 10 + 5h
Shop B: 7h
Set them equal: 10 + 5h = 7h
Step 3: Solve: h = 5
Answer: The two shops will cost the same after 5 hours.
Real-World Applications
These problems show up in many real-world situations:
- Business: Profit, loss, and pricing
- Transportation: Comparing routes and costs
- Budgeting: Setting limits and goals
- Scheduling: Managing time and resources
In each case, the first step is recognizing what kind of math can model the problem—whether it’s a linear equation (equality) or an inequality (limit or range).
Tips for Success
- Always label your variables clearly
- Write units (dollars, hours, students) in your final answers
- Check whether the solution needs to be a whole number (e.g., you can’t have half a student!)
- If the question includes phrases like “at most,” “no more than,” or “at least,” you’re likely dealing with an inequality
What You’ve Learned
By now, you’ve learned to:
- Translate word problems into linear equations and inequalities
- Use graphs and algebra to solve them
- Apply math to real-life scenarios
You’re building on skills from earlier topics like:
- Algebraic Expressions and Equations
- Graphing Linear Equations
- Graphing Linear Inequalities
- Slope and Rate of Change
Ready to take it further? In the next lesson, we’ll explore graphing systems of linear inequalities—situations where multiple conditions must be true at once.