🧮 Topic 1: Solving Linear Equations
What actually is a "linear equation"?
Think of a linear equation as a balance scale. Both sides are perfectly equal — that's what the "=" sign means. Your job isn't to change what's true, it's to peel away everything surrounding x until x is standing there completely alone, revealing what number it secretly was all along.
Formally: a linear equation can always be rearranged into the form ax + b = c, where a, b, and c are just numbers, and x is the one unknown. The word "linear" is the giveaway — it means x only ever appears to the power of 1. No x², no 1/x, no √x. If you tried to graph the relationship it would make a straight line — hence "linear."
Examples that count as linear: 2x + 3 = 5 3x + 4 = 1 x − 5 = −3
The golden rule: whatever you do to one side, do to the other
This is the entire chapter in one sentence. If you subtract 5 from the left side of the equation, you must subtract 5 from the right side too — otherwise the scale tips and the equation stops being true. Addition undoes subtraction (and vice versa); multiplication undoes division (and vice versa). These paired opposites are called inverse operations, and solving an equation is really just a game of "undo the operations in reverse order until x is alone."
Worked Example — Two-Step Equation
Solve 2x + 1 = 9
1Subtract 1 from both sides to undo the "+1": 2x = 8
2Divide both sides by 2 to undo the "×2": x = 4
Check it: substitute x = 4 back in → 2(4) + 1 = 9 ✓ It balances, so we're done.
Does the order of steps matter?
No — as long as each step is applied correctly, you can undo operations in any order. But some orders are much easier than others. For 4x + 8 = 12, it's far easier to subtract 8 first (giving 4x = 4, then x = 1) than to divide by 4 first. If you divide first, you must divide every single term by 4 — including the 8 — otherwise you'll wrongly get x + 8 = 3 instead of the correct x + 2 = 3.
Equations with negative numbers
Nothing new here conceptually — just be extra careful with signs when you divide. Remember: a negative divided by a negative gives a positive.
Worked Example
Solve 2 − 3x = 10
1Subtract 2 from both sides: −3x = 8
2Divide both sides by −3: x = −8/3
Some people prefer to rewrite 2 − 3x as −3x + 2 first (they're equivalent — addition can be written in any order) — then subtract 2 and divide by −3 exactly as before.
Practice Question 1
Solve the equation: 9 − 7x = 5
Equations with brackets
If brackets are in the way, your simplest move is almost always to expand them first, then solve as normal. (Technically you could divide both sides by the number outside the bracket instead — but that route often lands you with messier fractions, so expanding first is the safer default.)
Worked Example
Solve 5(3 − 4x) + 1 = 26
1Expand the bracket: 15 − 20x + 1 = 26
2Simplify the numbers on the left: 16 − 20x = 26
3Subtract 16 from both sides: −20x = 10
4Divide both sides by −20: x = −1/2 (or −0.5)
Equations with fractions
Fractions look scary, but there's a clean trick: multiply every single term on both sides by the lowest common denominator (LCD). This clears every fraction in one move, turning the equation back into the familiar whole-number kind you already know how to solve.
What if x is on the denominator?
Same idea, slightly different target: multiply both sides by whatever expression is on the bottom. For 4/(x−2) = 3, multiply both sides by (x − 2): this gives 4 = 3(x − 2). Expand and solve as normal: 4 = 3x − 6, so 3x = 10, and x = 10/3.
Practice Question 2
Solve the equation: 5x/4 = 1/2
Equations with x on both sides
When x shows up on both sides of the "=" sign, your very first job is to gather all the x-terms onto one side — before you touch anything else. The smart move is to remove the x-term from whichever side has the smaller coefficient (the smaller number in front of x), because that keeps you working with positive numbers and avoids extra sign headaches.
Worked Example
Solve 4 − 5x = 6x − 29
1−5x is smaller than 6x, so remove the x-term from the left by adding 5x to both sides: 4 = 11x − 29
2Add 29 to both sides to isolate the x-term: 33 = 11x
3Divide both sides by 11: x = 3
Practice Question 3
Solve the equation: 4x − 7 = 11 + x
📏 Topic 2: Solving Linear Inequalities
What is an inequality, really?
An equation says two things are exactly equal. An inequality is looser — it says one thing is bigger than or smaller than another, and instead of one single answer, there's a whole range of possible values that make it true.
x > 5
x is strictly greater than 5. So x could be 6, 7, 8, 9... or 5.001, or anything above 5 — but never 5 itself.
x < 5
x is strictly less than 5 — again, 5 itself is not allowed.
x ≥ 5 (greater than or equal to)
x is 5 or anything bigger. 5 is now included.
x ≤ 5 (less than or equal to)
x is 5 or anything smaller. 5 is included.
When the boundary value cannot be included (using > or <), we call it a strict inequality. When it can be included (≥ or ≤), it's non-strict. This distinction matters a lot — it's the difference between an open circle and a filled-in circle on a number line, and between including or excluding an integer in a list of solutions.
Finding integer solutions
Sometimes you're given two end points and asked to list every whole number (integer) that satisfies the inequality. The trick is simply to check: is each endpoint included or not?
3 ≤ x ≤ 6
Both ends included → x = 3, 4, 5, 6
3 ≤ x < 6
Left included, right excluded → x = 3, 4, 5
3 < x ≤ 6
Left excluded, right included → x = 4, 5, 6
3 < x < 6
Both ends excluded → x = 4, 5
Two classic traps
1) Zero and negative numbers count as integers too! x ≤ 2 gives x = 2, 1, 0, −1, −2, ... — don't forget to keep going below zero unless told "positive integers only."
2) If a question just says "x > 3" with no mention of integers, do not assume x has to be a whole number. It could be 3.1, or π (3.14159...), or anything above 3 at all. Only list integers if the question specifically asks for them.
You might also be asked to satisfy two inequalities at once — for example, 0 < x < 5 and x ≥ 3. List each one separately, then find the values that appear in both lists: {1, 2, 3, 4} and {3, 4, 5, 6, ...} overlap only at x = 3, 4.
Practice Question 4
List all the integer values of x that satisfy: −4 ≤ x < 2
Representing inequalities on a number line
A number line diagram turns an inequality into a picture: draw a circle above each end point, and join them with a horizontal line (or arrow, if there's only one end point).
If there's only one end point (like x > 5), draw a single circle at 5 and a horizontal arrow pointing in the direction of all the valid values — to the right for "greater than," to the left for "less than."
Quick visual check
For −3 < x ≤ 4: draw an open circle at −3 (excluded), a closed circle at 4 (included), and connect them with a line. Anything strictly between and including 4 is a valid solution.
Solving linear inequalities — the one rule that changes everything
Here's the good news: solving an inequality uses exactly the same steps as solving an equation — add, subtract, multiply, divide on both sides to isolate x. Here's the one rule that's genuinely new and easy to forget under exam pressure:
Why does this happen? Picture a number line. Multiplying by a negative number flips every value to the opposite side of zero — like a mirror. Whatever was "bigger" ends up smaller, and vice versa, so the inequality sign has to flip to stay true.
The safest strategy of all
Never multiply or divide by x itself — you don't know if x is positive or negative, so you can't know whether to flip the sign! The bulletproof approach: only ever add or subtract to move all the x-terms onto one side (the side that keeps the x coefficient positive). That way you avoid the flip rule altogether until the very last, safe division step.
Worked Example
Solve 2x − 5 ≤ 21
1Add 5 to both sides: 2x ≤ 26
2Divide both sides by 2 (positive, so no flip needed): x ≤ 13
Worked Example — x on both sides
Solve 10 − 2x > 4x − 7
1Add 2x to both sides (avoids a negative x-term): 10 > 6x − 7
2Add 7 to both sides: 17 > 6x
3Divide by 6 (positive — no flip): 17/6 > x, i.e. x < 17/6
Practice Question 5
Solve the inequality: 3 − 2x ≥ 11
Double (combined) inequalities
Sometimes x is sandwiched between two boundaries in one expression, like a < 2x < b. You solve this by applying the exact same operation to all three parts at once — the left side, the middle, and the right side.
🎯 Exam Tips & Common Mistakes
Always check your answer
Substitute your final value of x back into the original equation (before you did any rearranging). If both sides don't match, you've made an arithmetic slip somewhere — go back and find it before moving on.
Mistake: dividing only part of an expression
When dividing an equation like 4x + 8 = 12 by 4, you must divide every single term — including the 8 — not just the 4x. Writing x + 8 = 3 is a classic error that loses easy marks. Correct version: x + 2 = 3.
Mistake: forgetting to flip the inequality sign
This is the single most common inequality error on exam papers. Any time you multiply or divide both sides by a negative number, the sign must flip. Examiners specifically design questions to catch this — get in the habit of pausing and asking "am I dividing by a negative right now?" every time.
Mistake: turning an inequality into an equation
Never change the inequality sign to "=" partway through solving — even temporarily "to make it easier." This changes the entire meaning of the problem and will lose you marks in an exam, even if your arithmetic is otherwise correct.
Mistake: assuming x must be a whole number
Unless a question explicitly says "integer values," don't restrict your answer to whole numbers. x > 3 includes 3.5, 3.01, and even π — the full continuous range above 3, not just 4, 5, 6...
What examiners actually look for
Full marks usually require showing your working line by line — the operation you performed on each step (e.g. "(−5)" or "(÷2)") — not just the final answer. Method marks are awarded even if your final answer is wrong but your steps were logically correct, so never skip straight to the answer.
Typical exam question patterns for this chapter: "Solve the equation..." (with brackets/fractions/x on both sides), "List all integer values of x that satisfy...", "Represent this inequality on a number line", and "Solve the inequality..." (testing the flip rule).