Inequalities

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 meters 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. Don't worry if this seems tricky at first—once you know the "secret symbols," it’s just like solving regular equations!

1. What are Inequality Symbols?

An inequality tells us about the relationship between two values that are not necessarily equal. Here are the four main 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).

Memory Aid: The Hungry Crocodile

Imagine the inequality symbol is a hungry crocodile's mouth. The crocodile is very greedy and always wants to eat the biggest number! So, the "open" part of the mouth always points toward 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 \(x > 3\) has an infinite number of answers (like 4, 5, 6, and even 3.001), we use number lines to visualize them.

Open vs. Closed Circles

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

• Open Circle \(\circ\): Use this for \(<\) or \(>\). It means the number itself is not included. (Think of it as a hole that the number falls through!)
• Closed Circle \(\bullet\): Use this for \(\le\) or \(\ge\). It means the number is included. (The circle is "filled in" because the number stays there).

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 \(x > 2\), you would draw an open circle at 2 and an arrow pointing to the right.

Did you know?

If the letter \(x\) is on the left side of the symbol, the inequality sign actually looks like an arrow head pointing in the direction you need to draw! For example, in \(x > 5\), the symbol \(>\) points to the right \(\to\).

3. Solving Simple Inequalities

Solving an inequality is almost exactly like solving an equation. Your goal is to get the letter (the variable) all by itself on one side.

The Golden Rule

Whatever you do to one side of the inequality, you must do to the other side to keep it balanced.

Step-by-Step Example:

Solve: \(x + 4 < 10\)

1. We want \(x\) on its own. The \(+4\) is in the way.
2. Do the inverse (opposite): Subtract 4 from both sides.
3. \(x + 4 - 4 < 10 - 4\)
4. Answer: \(x < 6\)

Another Example:

Solve: \(3x \ge 12\)

1. The \(x\) is being multiplied by 3.
2. Do the inverse: Divide both sides by 3.
3. \(\frac{3x}{3} \ge \frac{12}{3}\)
4. Answer: \(x \ge 4\)

Key Takeaway: Use the "balance method" (doing the same thing to both sides) just like you do with equations!

4. Listing Integer Solutions

Sometimes, a question might ask you to list the integers that satisfy an inequality.
Reminder: An integer is just a whole number (it can be positive, negative, or zero).

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

Step 1: Look at the first part: \(2 < x\). This means \(x\) must be bigger than 2, so we cannot include 2.
Step 2: Look at the second part: \(x \le 6\). This means \(x\) can be smaller than 6 or exactly 6.
Step 3: Write down the whole numbers in between: 3, 4, 5, 6.
Final Answer: 3, 4, 5, 6.

5. Common Mistakes to Avoid

• Mistaking the symbols: Always double-check if the "mouth" is eating the right thing.
• Forgetting the "Or Equal To" line: Look closely at the symbol. If there is a line underneath (\(\le\)), your number line circle must be filled in!
• Losing the symbol: Some students accidentally turn the \(<\) or \(>\) into an \(=\) while they are working. Make sure your final answer is an inequality, not an equation!

Quick Review Box

Symbols: \(>\) (Greater), \(<\) (Less), \(\ge\) (Greater or equal), \(\le\) (Less or equal).
Circles: Open \(\circ\) for \(<\) and \(>\). Closed \(\bullet\) for \(\le\) and \(\ge\).
Solving: Use the balance method (do the same to both sides).
Integers: These are whole numbers. Check if the ends are included!

Great job! You’ve covered the basics of KS3 Inequalities. Keep practicing, and soon you'll be an expert at defining the boundaries of the math world!