Cambridge (CIE) IGCSE — International Maths: Extended
Linear Equations & Inequalities
Big idea: to solve for x, you get it alone on one side by doing the exact same operation — undoing it, really — to both sides of the equation (or expression), in whichever order makes the arithmetic easiest.
📋 Summary — What This Chapter Covers
- A linear equation is any equation you can rearrange into the form ax + b = c, where the highest power of x is just 1 (no x², no √x, nothing fancy).
- You solve equations by applying inverse operations to both sides until x stands alone — and the order you do this in doesn't matter, as long as you do it correctly.
- Equations with brackets get expanded first (or you can divide first — both work).
- Equations with fractions get cleared by multiplying every term by the lowest common denominator.
- Equations with x on both sides get solved by moving all the x-terms to one side first — pick the side with the smaller x-term to avoid extra negative signs.
- Inequalities (
>, <, ≥, ≤) are solved almost identically to equations — except when you multiply or divide both sides by a negative number, in which case you must flip the inequality sign.
- You can list the integer solutions of an inequality, represent inequalities on a number line (open circle = excluded, closed circle = included), and solve double/combined inequalities.
🧮 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?
4x + 8 = 12
every single term
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
1
2