Write and Solve an Inequality from Word Problems

📝 Practice Worksheets

Inequality Word Problems Writing and Solving Inequalities Writing One-Variable Inequalities Solving One-Variable Inequalities

🛠️ Related Tool

Inequality Graphing Explorer

🔑 Key Concepts

Many real-world problems involve a starting amount and a rate that changes over time.
Words such as maximum, no more than, and without exceeding indicate the symbol $\le$.
An inequality can represent how long something can continue before reaching a limit.
After writing the inequality, use inverse operations to isolate the variable.
The final answer should be interpreted in the context of the situation.

✏️ Worked Examples — Writing and Solving Inequalities with Limits

Example 1 — Saving Money

Problem:
Jordan has 250 dollars in savings and deposits 40 dollars each week. He wants to have no more than 650 dollars in the account. How many weeks can he continue depositing money?

Step 1: Define the variable.
Let $w$ represent the number of weeks.

Step 2: Write the inequality.
Starting amount + weekly deposit × weeks:

$250+40w\le650$

Step 3: Subtract $250$ from both sides.

$250+40w\ {\color{red}{-250}} \le 650\ {\color{red}{-250}}$

$40w\le400$

Step 4: Divide both sides by $40$.

$\displaystyle \frac{40w}{\color{red}{40}} \le \frac{400}{\color{red}{40}} $

$\boxed{w\le10}$

Interpretation:
Jordan can make deposits for up to 10 weeks without exceeding $650.

Example 2 — Text Messages

Problem:
A phone plan allows a maximum of 1,200 text messages per month. Maria has already sent 300 messages and sends about 25 messages each day. How many days can she continue texting without exceeding the limit?

Step 1: Define the variable.
Let $d$ represent the number of days.

Step 2: Write the inequality.
Starting messages + daily messages × days:

$300+25d\le1200$

Step 3: Subtract $300$ from both sides.

$300+25d\ {\color{red}{-300}} \le 1200\ {\color{red}{-300}}$

$25d\le900$

Step 4: Divide both sides by $25$.

$\displaystyle \frac{25d}{\color{red}{25}} \le \frac{900}{\color{red}{25}} $

$\boxed{d\le36}$

Interpretation:
Maria can continue texting for up to 36 days without exceeding the limit.

Example 3 — Filling a Pool

Problem:
A pool can hold a maximum of 4,800 gallons of water. It already contains 1,800 gallons. Water is added at a rate of 30 gallons per minute. How many minutes can water be added without exceeding the maximum capacity?

Step 1: Define the variable.
Let $m$ represent the number of minutes.

Step 2: Write the inequality.
Starting water + rate × minutes:

$1800+30m\le4800$

Step 3: Subtract $1800$ from both sides.

$1800+30m\ {\color{red}{-1800}} \le 4800\ {\color{red}{-1800}}$

$30m\le3000$

Step 4: Divide both sides by $30$.

$\displaystyle \frac{30m}{\color{red}{30}} \le \frac{3000}{\color{red}{30}} $

$\boxed{m\le100}$

Interpretation:
Water can be added for up to 100 minutes without exceeding the pool’s capacity.

🧠 Math Vocabulary

Maximum: the greatest amount allowed.
Rate: how fast a quantity changes over time.
Without Exceeding: means the value must stay less than or equal to a limit.
Inequality: a mathematical statement that compares two values.
Variable: a letter that represents an unknown quantity.

💡 Main Idea

Many real-world situations involve limits. When there is a starting amount and a constant rate of change, an inequality can determine how long an activity can continue before reaching a maximum or minimum value. Writing and solving the inequality helps model these real-life constraints.

📚 What You Should Already Know

Students should already know how to solve one-step equations, use inverse operations, and interpret inequality symbols such as $\le$ and $\ge$. Understanding how variables represent unknown quantities is also important before working with inequality word problems.

🚀 What Comes Next

After solving inequalities that involve limits, students will move on to graphing inequalities on a number line, solving multi-step inequalities, and applying inequalities to real-world decision making and problem solving.

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