Welcome to the World of Inequalities!

In Mathematics, we often spend a lot of time looking for the exact answer using the equals sign (\(=\)). But in the real world, things aren't always exactly equal! For example, you might need to be at least 1.2 metres tall to go on a rollercoaster, or your phone battery might need to be more than 5% to stay on.

In this chapter, we will learn how to describe these ranges of values using Inequalities. Once you know the symbols and conventions, you will be able to show ranges clearly on number lines and describe limits and error intervals!

1. What are Inequality Symbols?

An inequality tells us about the relationship between two values that are not necessarily equal. Here are the key symbols you need to know:

\(>\) Greater Than: The value on the left is bigger than the value on the right.
Example: \(10 > 2\) (Ten is greater than two).

\(<\) Less Than: The value on the left is smaller than the value on the right.
Example: \(3 < 8\) (Three is less than eight).

\(\ge\) Greater Than or Equal To: The value on the left is either bigger or exactly the same as the value on the right.
Example: \(x \ge 5\) (\(x\) could be 5, 6, 7, or even 5.1).

\(\le\) Less Than or Equal To: The value on the left is either smaller or exactly the same as the value on the right.
Example: \(x \le 10\) (\(x\) could be 10, 9, 8, or 9.9).

\(\ne\) Not Equal To: The value on the left is simply not equal to the value on the right.
Example: \(x \ne 0\) (\(x\) can be any number except zero).

Memory Aid: The Hungry Crocodile

Imagine the inequality symbol is a hungry crocodile's mouth. The crocodile always wants to eat the biggest number! So, the wide open part of the mouth always points towards the larger value.

Quick Takeaway: Inequalities show a range of possible answers, not just one fixed number.

2. Showing Inequalities on a Number Line

Since an inequality like \(x > 3\) includes an infinite set of numbers (such as 4, 5, 6, and even 3.01), we use number lines to visualise them.

Open vs. Closed Circles

This is the most important rule to remember when drawing your diagrams:

  • Open Circle \(\circ\): Use this for \(<\) or \(>\). It means the endpoint itself is not included in the range.
  • Closed Circle \(\bullet\): Use this for \(\le\) or \(\ge\). It means the endpoint is included.

Drawing the Arrow

  • If \(x\) is greater (\(>\) or \(\ge\)), draw the arrow pointing to the right (where numbers get bigger).
  • If \(x\) is less (\(<\) or \(\le\)), draw the arrow pointing to the left (where numbers get smaller).

Example: To show \(x > 2\), draw an open circle at 2 and an arrow pointing to the right.

Double Inequalities on a Number Line

When \(x\) is trapped between two values, such as \(1 \le x < 5\), draw a circle at each end and join them with a straight line:

  • A closed circle at 1 (because \(x \ge 1\)).
  • An open circle at 5 (because \(x < 5\)).
  • A solid line connecting the two circles.

3. Double Inequalities and Error Intervals

An inequality can show that a number lies between a lower limit and an upper limit. We write this as a double inequality, placing the variable in the middle.

Example: \(4 < x \le 9\) means that \(x\) is strictly greater than 4, and less than or equal to 9.

Error Intervals from Rounding

When values are rounded to estimate answers, we can describe the possible resulting errors using inequality notation.

Worked Example: A length \(L\) is measured as \(8\text{ cm}\), rounded to the nearest centimetre. What is the error interval for \(L\)?

1. The value could be as small as \(7.5\text{ cm}\) (which rounds up to 8).
2. The value must be strictly less than \(8.5\text{ cm}\) (because 8.5 would round up to 9).
3. Using inequality notation, we write the error interval as:
\(7.5 \le L < 8.5\)

This standard notation of the form \(a \le x < b\) (or \(a < x \le b\)) clearly defines the minimum and maximum possible values of the original measure.

4. Listing Integer Solutions

Sometimes, a question asks you to list the integers that satisfy an inequality.
Reminder: An integer is a whole number (positive, negative, or zero).

Example Question: List all the integers where \(2 < x \le 6\).

Step 1: Look at the lower bound: \(2 < x\). Since \(x\) must be strictly greater than 2, we cannot include 2.
Step 2: Look at the upper bound: \(x \le 6\). Since the symbol is \(\le\), \(x\) can be smaller than 6 or exactly 6.
Step 3: List the whole numbers in between: 3, 4, 5, 6.
Final Answer: 3, 4, 5, 6.

5. Common Mistakes to Avoid

  • Confusing the symbols: Always double-check which side the open mouth points towards.
  • Forgetting whether to shade circles: If there is an "or equal to" bar (\(\le\) or \(\ge\)), the circle must be filled in (\(\bullet\)). If it is a strict inequality (\(<\) or \(>\)), leave it open (\(\circ\)).
  • Including excluded endpoints when listing integers: In \(-1 < x \le 3\), \(-1\) is not included because the symbol is \(<\), so the integer list begins at 0.

Quick Review Box

Symbols: \(>\) (greater than), \(<\) (less than), \(\ge\) (greater than or equal to), \(\le\) (less than or equal to), \(\ne\) (not equal to).
Number Lines: Open circle \(\circ\) for strict inequalities (\(<, >\)); closed circle \(\bullet\) for inclusive inequalities (\(\le, \ge\)).
Double Inequalities: Place \(x\) between two values to show bounds (e.g. \(a < x \le b\)).
Error Intervals: Use inequalities to state the range of possible original values after rounding (e.g. \(7.5 \le L < 8.5\)).
Integers: Whole numbers satisfying the range. Always verify whether the endpoints are included!

Great job! You now understand the foundations of Key Stage 3 inequalities, from number lines to error intervals.