Chapter One of Inequalities: Understanding and Writing Inequalities

1.1 What is an inequality?

An inequality is a mathematical sentence that compares two values or expressions using symbols that show a relationship of inequality, rather than equality.

We use inequalities to describe situations where one value is greater than, less than, greater than or equal to, or less than or equal to another.

1.2 Identifying Inequalities

Let’s start by identifying whether a given expression is an inequality.

Example:

What kind of mathematical sentence is this:
x > 27 - 6?

• It is not an equation because there is no equal sign.
  
  

• It is not just an expression, because it includes a comparison.
  
  

• The symbol > tells us that it's an inequality.
  
  

✅ Answer: It is an inequality.

1.3 Writing Basic Inequalities

To write an inequality, you need to:

1. Understand the relationship described in the sentence.
   
   

2. Choose the correct symbol based on the words used.
   
   

3. Translate it into a mathematical expression.
   
   

Example:

Write an inequality for:
“𝑐 is less than or equal to 3”

✅ The phrase “less than or equal to” means use the symbol ≤.

📘 Inequality: c ≤ 3

1.4 Real-World Situations with Inequalities

In real life, inequalities help us describe limits, maximums, minimums, or conditions that aren’t exact but fall within a range.

Example 1: Subscription Fee

The subscription fee is no more than $74.

✅ “No more than” means the value can be equal to or less than 74.

📘 Inequality: x ≤ 74

Example 2: Race Time Comparison

Sameh finished a race in 90 seconds. Sally finished at least 9 seconds earlier. What inequality represents Sally's time?

✅ "At least 9 seconds earlier" means Sally took 90 - 9 = 81 seconds or less.

📘 Let 𝑥 be Sally’s time:
x ≤ 81

Example 3: Planning a Journey

Yara walks 7 minutes to the bus stop. The bus comes every 12 minutes, takes 20 minutes to reach the station, and she takes at most 4 minutes to reach the platform. Write an inequality for the total time, 𝑡.

🧠 Add the maximum values:
7 (walk) + 12 (wait) + 20 (ride) + 4 (walk) = 43 minutes

📘 Inequality: t ≤ 43

If the train leaves at 14:47, when should she leave home?

🚶‍♀️ She should leave by 14:47 - 43 minutes = 14:04

✅ Answer: Before 14:04

1.5 Practice Problems

Try writing an inequality for each sentence below:

1. A box can hold no more than 12 books.
   ✅ ___
   
   

2. You must be at least 16 years old to drive.
   ✅ ___
   
   

3. Sam got more than 80 points in the quiz.
   ✅ ___
   
   

4. The room can hold a maximum of 25 people.
   ✅ ___
   
   

5. You need to run less than 5 miles today.
   ✅ ___
   
   

Answers:

1. 𝑏 ≤ 12
   
   

2. 𝑎 ≥ 16
   
   

3. 𝑠 > 80
   
   

4. 𝑟 ≤ 25
   
   

5. 𝑚 < 5
   
   

1.6 Summary

✅ Key Takeaways:

• Inequalities compare two values using <, >, ≤, or ≥.
  
  

• Inequalities often appear in real-life situations involving limits or conditions.
  
  

• Verbal expressions like “no more than” or “at least” can be translated into inequalities.
  
  

• Diagrams or timelines can help visualize complex scenarios.
  
  

• Writing an inequality involves identifying the relationship and using the correct symbol.
  
  

In the next chapter, we will learn how to graph inequalities on a number line, which will help us visualize solutions and better understand the range of possible values for a variable.