Mathematics (Specification A) | Inequalities

Welcome to the World of Inequalities!

Hello future mathematicians! In the world of "Equations, formulae and identities," we usually focus on finding an exact answer, like \(x = 5\). But what happens when the answer isn't just one number? That's where Inequalities come in!

An inequality tells us that one value is not equal to another, but instead, it is greater than or less than the other. Think of it like a speed limit sign: it doesn't say "drive exactly 50 mph," it says "drive 50 mph or less!"

In this chapter, we will learn how to read, solve, and represent these ranges of solutions, ensuring you are ready to tackle any IGCSE question on the topic!

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Section 1: The Language of Inequalities

Before we start solving, we need to understand the four key symbols. Think of these as the mathematical punctuation for expressing comparisons.

The Four Key Inequality Symbols

We use the following symbols to show relationships between numbers or expressions:

1. Less Than (\(<\))
Example: \(x < 7\) means \(x\) could be 6, 0, or -10, but it cannot be 7.

2. Greater Than (\(>\))
Example: \(y > -2\) means \(y\) could be -1, 0, or 100, but it cannot be -2.

3. Less Than or Equal To (\(\leq\))
This is often called "inclusive."
Example: \(P \leq 10\) means \(P\) can be 10 or any number smaller than 10. The number 10 is included in the solution.

4. Greater Than or Equal To (\(\geq\))
This is also "inclusive."
Example: \(Q \geq 5\) means \(Q\) can be 5 or any number larger than 5. The number 5 is included in the solution.

Memory Aid: How to remember which way the sign points?

The inequality sign always points to the smaller number, and the open mouth (the larger side) always faces the bigger number.
Analogy: Imagine the sign is a hungry crocodile; it always wants to eat the biggest meal!

Key Takeaway from Section 1:

Inequalities represent a range of possible solutions. Remember whether the sign includes the boundary number (using \(\leq\) or \(\geq\)) or excludes it (using \(<\) or \(>\)).

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Section 2: Representing Inequalities on a Number Line

The easiest way to visualise an inequality is by drawing it on a number line. This helps us see the entire range of possible solutions.

Hollow vs. Solid Circles (The Boundary Markers)

When drawing an inequality, we use a circle to mark the boundary point (the number itself). The type of circle tells us whether the boundary point is included in the solution set.

1. The Hollow Circle (O)

Use a hollow (open) circle when the number is NOT included.
This is used for \(<\) (Less Than) and \(>\) (Greater Than).
Example: For \(x > 3\), you draw a hollow circle over 3.

2. The Solid Circle (•)

Use a solid (filled-in) circle when the number IS included.
This is used for \(\leq\) (Less Than or Equal To) and \(\geq\) (Greater Than or Equal To).
Example: For \(x \leq 3\), you draw a solid circle over 3.

Step-by-Step: Drawing \(x \geq -1\)

1. Identify the boundary: The number is -1.
2. Choose the circle type: Since it is \(\geq\), it is "Greater Than or Equal To," so we use a solid dot (•).
3. Determine the direction: We need numbers "greater than" -1. Numbers get larger as you move to the right on a number line.
4. Draw the line: Draw a thick line starting at the solid dot on -1 and extending to the right, often with an arrow at the end to show it continues forever.

Quick Tip: For variables written on the left (e.g., \(x < 5\)), the direction of the arrow you draw usually matches the direction of the inequality sign itself (<\(\rightarrow\) Left; > \(\rightarrow\) Right). Be careful, though! This trick only works if \(x\) is on the left.

Key Takeaway from Section 2:

Open circles mean Exclusive (\(<\) or \(>\)). Solid circles mean Inclusive (\(\leq\) or \(\geq\)). The arrow shows the direction of all the possible solutions.

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Section 3: Solving Linear Inequalities (The Golden Rule)

Don't worry! Solving linear inequalities is almost identical to solving linear equations. You use the same operations (add, subtract, multiply, divide) to isolate the variable \(x\).

The Process – Same as Equations

To solve an inequality, you must keep the expression balanced. Whatever you do to one side of the inequality, you must do to the other side.

Example 1: Basic Addition/Subtraction

Solve \(x - 5 < 12\).

Step 1: We need to isolate \(x\). Add 5 to both sides.
\(x - 5 + 5 < 12 + 5\)

Step 2: Simplify.
\(x < 17\)

Example 2: Basic Multiplication/Division (Positive Numbers)

Solve \(3x \leq 21\).

Step 1: Divide both sides by 3.
\(\frac{3x}{3} \leq \frac{21}{3}\)

Step 2: Simplify.
\(x \leq 7\)

The Golden Rule of Inequalities: Flipping the Sign

This is the ONE difference between equations and inequalities, and it is the most common place to lose marks.

If you multiply or divide the inequality by a negative number, you MUST reverse (flip) the direction of the inequality sign.

Why does this happen? (Did you know?)

Look at two numbers: \(2 < 5\). This is true.
If we multiply both sides by \(-1\):
\(2 \times (-1) = -2\) and \(5 \times (-1) = -5\).
On a number line, \(-2\) is bigger than \(-5\). Therefore, the relationship must flip: \(-2 > -5\).

Example 3: Applying the Golden Rule

Solve \(-4x > 8\).

Step 1: Divide both sides by -4.
\(\frac{-4x}{-4} \quad \frac{8}{-4}\)

Step 2: Because we divided by a negative number (\(-4\)), we must flip the sign from \(>\) to \(<\).
\(x < -2\)

Key Takeaway from Section 3:

Solve inequalities like equations, but remember the Golden Rule: flip the sign if you multiply or divide by a negative number.

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Section 4: Double the Fun – Compound Inequalities

Sometimes a variable is restricted by two conditions at once. These are called compound or double inequalities. They look like this:
\(-3 \leq 2x + 1 < 7\)

This inequality means that \(2x + 1\) must be greater than or equal to -3 AND less than 7 at the same time.

Solving Compound Inequalities

The key is to treat the inequality as three parts: the left side, the middle, and the right side. You must perform the same operation on all three parts to isolate \(x\) in the middle.

Example 4: Solving a Compound Inequality

Solve \(-3 \leq 2x + 1 < 7\).

Step 1: Get rid of the +1 in the middle.
Subtract 1 from all three sections:
\(-3 - 1 \leq 2x + 1 - 1 < 7 - 1\)

Step 2: Simplify.
\(-4 \leq 2x < 6\)

Step 3: Isolate \(x\) by dividing by 2.
(Since 2 is positive, we DO NOT flip the signs).
\(\frac{-4}{2} \leq \frac{2x}{2} < \frac{6}{2}\)

Step 4: Simplify to find the solution.
\(-2 \leq x < 3\)

This solution means \(x\) is any number between -2 (inclusive) and 3 (exclusive).

Plotting the Compound Inequality \(-2 \leq x < 3\)

1. Mark -2 with a solid dot (due to \(\leq\)). 2. Mark 3 with a hollow dot (due to \(<\)). 3. Draw a line connecting the two dots.

Key Takeaway from Section 4:

Compound inequalities are solved by applying the same operation to the left, middle, and right parts simultaneously, ensuring \(x\) is isolated in the middle.

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Section 5: Finding Integer Solutions

Often in IGCSE questions, after you have solved an inequality, you will be asked to list the integer solutions.

An integer is a whole number (positive, negative, or zero). It cannot be a fraction or a decimal.

Example 5: Finding Integers

Find the integer solutions for \( -2 \leq x < 3 \).

1. The solution starts at -2 and includes it (\(\leq\)). 2. The solution goes up to numbers less than 3, meaning 3 itself is not included.

The whole numbers that fit this range are: -2, -1, 0, 1, and 2.

Example 6: Integer Solutions (Revisiting a Solution)

Suppose you solved an inequality and found the solution to be \(x > 4.5\).

What are the smallest three integers that satisfy this?

\(x\) must be greater than 4.5. The next whole number is 5.
The smallest three integers are: 5, 6, and 7.

Encouragement: Finding the integer solution is the final step where you translate your abstract mathematical answer back into a practical list. Always check your sign to make sure you include (or exclude) the boundary number correctly!

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Quick Review Checklist

Inequalities

• Symbols: \(<\), \(>\) (Exclusive) | \(\leq\), \(\geq\) (Inclusive)
• Number Line: Open Circle for Exclusive, Solid Circle for Inclusive.
• Solving: Same steps as equations (isolate \(x\)).
• Golden Rule: Flip the sign if multiplying or dividing by a negative number.
• Compound: Work on all three sections simultaneously.
• Integer Solutions: List the whole numbers (0, 1, 2, -1, -2, etc.) that fall within the solved range.

You've successfully covered the core concepts of linear inequalities! Practice makes perfect, especially the crucial "Golden Rule"!