Algebra: All Connected
These topics are not separate chapters. They are one system where each concept unlocks the next. Start here and work through in order.
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Book Now →Before you can work with expressions, equations or exponents, you need to know what kind of number you are dealing with. The real number system is the map - it tells you where every number lives and what rules apply to it.
| Set | Symbol | What it includes | Examples |
|---|---|---|---|
| Natural Numbers | ℕ | Counting numbers, starting from 1 | 1, 2, 3, 4, 5 ... |
| Whole Numbers | ℕ₀ | Natural numbers PLUS zero | 0, 1, 2, 3, 4 ... |
| Integers | ℤ | Whole numbers AND their negatives | ... -3, -2, -1, 0, 1, 2, 3 ... |
| Rational Numbers | ℚ | Any number that can be written as a fraction a/b (b ≠ 0). Includes all terminating and recurring decimals. | ½, -3, 0.75, 0.̅3̅, 2⅓ |
| Irrational Numbers | ℚ′ | Cannot be written as a fraction. Non-terminating, non-recurring decimals. | √2, π, √5 |
| Real Numbers | ℝ | ALL of the above combined | Every number on the number line |
The Nesting Rule - Sets inside Sets
Given: √16, √3, -7, 0.121212..., π. Answers: √16 = 4 (rational/integer), √3 (irrational), -7 (integer/rational), 0.12̅ (rational - it recurs), π (irrational). Always simplify surds first before classifying.
A number line puts every real number in its correct position. Once you can place any number on a line, you can order numbers from smallest to largest, solve inequalities, and understand which values a variable can take. It is also the tool you use to show the answer to an inequality.
Every number has exactly one position on the real number line. Numbers to the RIGHT are always LARGER. Numbers to the LEFT are always SMALLER. Zero sits in the middle. Negative numbers go left, positive numbers go right.
Placing Different Types of Numbers
Visualising the ordering above
To place √n on a number line: find the two consecutive integers a and b where a² < n < b². Then √n lies between a and b. For a closer estimate, check whether n is nearer to a² or b². For example √11: 3² = 9 and 4² = 16, so √11 is between 3 and 4. Since 11 is closer to 9 than to 16, it sits closer to 3 than to 4.
Students forget that negative numbers reverse the size comparison. -3 is SMALLER than -1, even though 3 is bigger than 1. On the number line, -3 is further LEFT. Always check direction before ordering negative numbers.
Exponents appear in algebraic expressions, equations and factorisation. You need to be comfortable simplifying a⁴ x a² before you can work with expressions confidently. Learn these rules cold.
Multiply - Same Base
Add the exponents. Keep the base.
Divide - Same Base
Subtract the exponents. Keep the base.
Power of a Power
Multiply the exponents.
Zero Exponent
Any base (except 0) raised to the power zero equals 1.
Negative Exponent
Flip to the denominator and make the exponent positive.
Fraction Exponent
A fraction exponent is the same as a root.
Power of a Product
The exponent applies to ALL factors inside the bracket.
Power of a Fraction
Apply the exponent to both numerator and denominator.
The Common Mistakes
Simplifying expressions, factorising, solving equations involving powers - all of it uses these eight laws. A student who knows the exponent laws fluently will find the rest of algebra much more manageable.
Prime Base Conversion
When bases like 4, 8, 9, 25 appear, rewrite them as prime numbers raised to a power FIRST. Then apply the exponent laws. This is the only way to combine terms that look different but share a prime base.
Never try to combine 4ⁿ and 8ⁿ directly. Rewrite as powers of 2 first, every time. If you cannot see how to simplify, ask yourself: what prime number does this base reduce to?
Adding and Subtracting Exponential Expressions
The multiplication and division exponent laws ONLY work when terms are multiplied or divided. When terms are ADDED or SUBTRACTED you cannot apply those laws. Instead, factor out the lowest power as a common factor, or use the k-substitution method.
2³ + 2² ≠ 2⁵. The law aᵏ x aᵐ = aᵏ⁺ᵐ only applies to MULTIPLICATION. When you add or subtract, you must treat the terms differently: factor out the lowest power as a common factor.
Method 1: Factor out the lowest power
Method 2: k-Substitution
2³ + 2² is NOT 2⁵. The answer is 8 + 4 = 12. The exponent laws only work under multiplication and division. When you see a + or - between exponential terms, factor out the common lowest power or use k-substitution.
An equation says two expressions are equal. Before you can solve it, you need to be able to simplify and manipulate expressions on each side. Think of expressions as the "ingredients" and equations as the "recipe" - you need to understand the ingredients first.
Term: a single part of an expression (e.g. 3x², -5y, 7). Coefficient: the number in front of the variable (in 3x², the coefficient is 3). Like terms: same variable AND same exponent (3x² and -7x² are like terms; 3x² and 3x are NOT).
Collecting Like Terms
3x² + 5x - 2x² + x = (3x² - 2x²) + (5x + x) = x² + 6x
Multiplying Expressions - Three Types
(x + 3)² ≠ x² + 9. You MUST expand it: (x + 3)(x + 3) = x² + 6x + 9. The middle term 2ab always gets left out by students who try to shortcut it. Learn the pattern or always expand it in full.
Factorisation is the key that unlocks both. To solve x² - 5x + 6 = 0, you need to factorise it. To simplify (x² - 9)/(x + 3), you need to factorise the numerator. Without factorisation, both of those are stuck. This is one of the most important skills in Grade 9 algebra.
The Order to Always Follow - Try Each Method in This Sequence
Common Factor First - Always
Before anything else, check if every term shares a common factor. Take it out.
6x² + 9x = 3x(2x + 3)
Difference of Two Squares
Two perfect squares separated by a minus sign.
a² - b² = (a + b)(a - b)
x² - 25 = (x + 5)(x - 5) | 4x² - 9 = (2x + 3)(2x - 3)
Trinomial (Quadratic): x² + bx + c
Find two numbers that MULTIPLY to give c and ADD to give b.
x² + 5x + 6: find two numbers that multiply to 6, add to 5 → 2 and 3 → (x + 2)(x + 3)
Grouping (4 Terms)
Group into two pairs and take a common factor from each pair.
ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)
Signs Pattern for Trinomials
Students who jump straight to trinomial factorisation without checking for a common factor first always get stuck. 2x² + 6x + 4 looks like a hard trinomial but pull out 2 first: 2(x² + 3x + 2) = 2(x + 1)(x + 2). Common factor first, every single time.
Grouping with Sign Change
Sometimes grouping does not immediately reveal a common bracket. When the signs of the second pair work against you, factor out a NEGATIVE from that pair to force the brackets to match.
After taking a common factor from the first pair, if the remaining bracket does NOT match the second pair, try factoring out a negative from the second pair. A negative flips all the signs inside, which often creates a matching bracket.
If you get (x - 3) from the first pair and (3 - x) from the second pair, these are NOT the same - but they are negatives of each other. Factor -1 from (3 - x) to get -(x - 3), and then the brackets match. This is the most commonly missed technique in grouping questions.
To simplify (x² - 9)/(x + 3), you must first factorise the numerator into (x + 3)(x - 3), then cancel the (x + 3) that appears in both. Without factorisation, you cannot simplify algebraic fractions. They are directly linked.
Simplifying - Factorise First, Then Cancel
(x² - 9)/(x + 3) = (x + 3)(x - 3)/(x + 3) = (x - 3). Cancel the common factor (x + 3).
Multiplying Fractions - Multiply Straight Across
(a/b) x (c/d) = ac/bd. Factorise first, cancel common factors, then multiply.
Dividing Fractions - Flip the Second and Multiply
(a/b) ÷ (c/d) = (a/b) x (d/c) = ad/bc. Keep, change, flip.
Adding and Subtracting - LCD First
Find the Lowest Common Denominator. Rewrite each fraction with the LCD. Then add/subtract numerators only. Example: (1/x) + (2/x²): LCD = x². Rewrite: (x/x²) + (2/x²) = (x + 2)/x².
The Rules
(x + 5)/5 does NOT simplify to x + 1. You cannot cancel the 5 into (x + 5) because 5 is not a factor of (x + 5) - it is only part of it. Only cancel when the same expression appears as a complete factor in both numerator and denominator.
Whatever you do to one side of an equation, you MUST do to the other. The equation is a balance - keep it balanced at every step. Your goal is always to get the variable alone on one side.
Linear Equations (one step at a time)
Quadratic Equations - THIS is why you needed factorisation
x² = 5x is NOT ready to factorise. Move everything to one side first: x² - 5x = 0, then x(x - 5) = 0, so x = 0 or x = 5. Students who leave it as x² = 5x and then divide both sides by x lose the solution x = 0 and lose marks.
Word Problems and Modelling
A word problem asks you to translate English into algebra. The equation is always hidden inside the words. Follow the same five steps every time.
Never start a word problem by writing an equation first. Write "Let x = ..." clearly at the top. If the question involves two unknowns (like two ages), express both in terms of x. Examiners award marks for the definition, the equation, and the solution - all three must be visible.
An inequality is solved exactly like a linear equation with one extra rule. If you can solve equations (Topic 6), you can solve inequalities. The only new element is: what does the answer look like on a number line, and what happens when you multiply or divide by a negative number.
Less than
Open circle on number line. That value is NOT included.
Less than or equal
Closed (filled) circle. That value IS included.
Greater than
Open circle. Arrow points right. Value NOT included.
Greater than or equal
Closed circle. Arrow points right. Value IS included.
Solving - Same as Equations with One Critical Difference
-2x < 8 → divide by -2 → x > -4 (sign flipped)
Number Line: x < 4 (open circle) vs x ≤ 4 (closed circle)
You have now covered the complete algebra system in order. Real numbers gave you the foundation. Exponents gave you the tools. Expressions gave you the language. Factorisation gave you the key. Fractions applied that key. Equations used all of it. Inequalities extended equations with one new rule. None of these topics stands alone - they are one connected system.
Quick Reference
Rules to Never Forget
Exponents
Same base: add exponents (multiply), subtract (divide), multiply (power of power).
Factorisation Order
Common factor → Diff of squares → Trinomial → Grouping
Quadratic Equations
= 0 first. Factorise. Zero product rule. Two solutions.
Inequality Flip
Multiply or divide by a NEGATIVE → sign flips. Always.
Algebraic Fractions
Factorise first. Cancel complete factors only. State restrictions.
Special Products
(a+b)² = a²+2ab+b². (a+b)(a-b) = a²-b².
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