Linear Equations and Inequalities
Solving equations is the most widely used skill in the whole syllabus — it turns up inside geometry, trigonometry, ratio and statistics questions. Inequalities work almost identically, with one rule that changes everything and a set of conventions about circles, lines and shading that carry marks on their own.
1. Solving linear equations
Whatever you do to one side, do to the other. That keeps the equation balanced.
Example: 4x + 5 = 21
- Subtract 5 from both sides: 4x = 16
- Divide both sides by 4: x = 4
Example: 4x = 18
- x = 18 ÷ 4 = 4.5
A fraction or decimal answer is not a sign you’re wrong. Leaving 4x = 18 as “x = 18” because 18 doesn’t divide neatly was a recorded error. Divide anyway and give 4.5 or 9/2.
“Moving a term across and changing the sign” is a shortcut for subtracting it from both sides. It works, but if you are making sign errors, write out the full operation on both sides — that habit fixes most of them.
With brackets
Example: 3(x + 4) = 27
- Expand: 3x + 12 = 27
- 3x = 15
- x = 5
Multiply the bracket by everything inside it. 3(x + 4) is 3x + 12, not 3x + 4.
With unknowns on both sides
Example: 5x − 3 = 2x + 12
- Subtract 2x from both sides: 3x − 3 = 12
- Add 3: 3x = 15
- x = 5
Collect the letters on the side that keeps them positive. Here, moving 2x left rather than 5x right avoids a negative coefficient and the sign errors that follow it.
With fractions
Example: (x + 3)/4 = 5
Multiply both sides by 4: x + 3 = 20 → x = 17
Example: x/2 + x/3 = 10
Multiply every term by 6 (the LCM): 3x + 2x = 60 → 5x = 60 → x = 12
Multiply EVERY term by the denominator, including terms that don’t have a fraction in them. Missing one is the standard error here.
Checking
Substitute your answer back into the original equation. For 5x − 3 = 2x + 12 with x = 5: left = 22, right = 22. ✓
2. Forming equations from words
- Define the unknown: “let the number of red counters be x”
- Translate each sentence into algebra
- Solve
- Answer the question that was asked — sometimes x is not the final answer
Check what the question actually wants. If x is the smaller number and the question asks for the larger, one more step is needed. Marks are regularly lost by stopping at x.
3. Inequalities
| Symbol | Meaning | On a number line |
|---|---|---|
| < | less than | open circle ○ |
| > | greater than | open circle ○ |
| ≤ | less than or equal to | closed circle ● |
| ≥ | greater than or equal to | closed circle ● |
The circle shows whether the boundary value is included. ≤ and ≥ include it (filled in); < and > exclude it (hollow). Getting this wrong is a whole mark, and it was one of the most frequent errors recorded in lessons.
Solving them
Solve exactly as you would an equation — with one exception:
When you multiply or divide by a NEGATIVE number, REVERSE the inequality sign.
Example: −2x > 6
- Divide both sides by −2, and flip: x < −3
Check it: x = −4 satisfies x < −3, and −2(−4) = 8 > 6 ✓. Without the flip you would get x > −3, which x = 0 would satisfy — but −2(0) = 0, which is not greater than 6. ✗
Only for multiplying or dividing by a negative. Adding or subtracting a negative number does not flip the sign.
A way to avoid it entirely: keep the letter positive from the start. For −2x > 6, add 2x to both sides and subtract 6: −6 > 2x, so 2x < −6, giving x < −3 with no flip needed.
Example: 5 − 3x ≤ 14
- −3x ≤ 9
- Divide by −3 and flip: x ≥ −3
Double inequalities
Example: −3 < 2x + 1 ≤ 9
Do the same thing to all three parts:
- subtract 1: −4 < 2x ≤ 8
- divide by 2: −2 < x ≤ 4
Integer solutions
“List the integers satisfying −2 < x ≤ 4” → −1, 0, 1, 2, 3, 4
Check both ends carefully. −2 is excluded (strict <) but 4 is included (≤). Wrongly including or excluding an endpoint was a repeated error, and it is the entire point of the question.
Give an inequality, not an equation, as your answer. Writing “x = −3” when the answer is “x ≥ −3” loses the mark.
4. Regions on a graph
Harder questions ask you to show a region satisfying several inequalities.
Method:
- Replace each inequality sign with = and draw that line
- Decide solid or dashed
- Decide which side to shade
- Label the region
Dashed line for < or > (the boundary is not included). Solid line for ≤ or ≥ (the boundary is included). This mirrors the open/closed circle rule on a number line.
Choosing the side — the reliable test: pick a point not on the line, usually (0, 0), and substitute it into the inequality. If it is true, shade the side containing that point; if false, shade the other side.
Example: y ≥ 2x + 1. Test (0,0): is 0 ≥ 1? No. So shade the side not containing the origin.
Read whether the question wants the required region shaded or unshaded. Some papers ask you to shade the region that does not satisfy the inequalities, leaving the answer region clear. Both conventions appear, and students lost marks by assuming. Label your region — usually R — so there is no doubt.
Common boundary lines: x = a is vertical; y = b is horizontal; y = x is the diagonal through the origin.
5. Mistakes that cost marks
Not doing the same thing to both sides.
Sign errors when moving terms across.
Expanding a bracket onto only the first term.
Multiplying only some terms when clearing fractions.
Stopping at x when the question asked for something else.
Forgetting to flip the inequality when multiplying or dividing by a negative.
Flipping when adding or subtracting, which is not required.
Using the wrong circle on a number line.
Including or excluding the wrong endpoint in a list of integers.
Writing an equation instead of an inequality.
Solid where it should be dashed, or vice versa.
Shading the wrong side — or the wrong one of “required”/“not required”.
Rejecting a fractional answer as if it must be wrong.
Frequently asked questions
How do you solve a linear equation? Do the same operation to both sides until the letter is alone.
What do I do with brackets? Expand them first, multiplying everything inside by the term outside.
How do I deal with fractions in an equation? Multiply every term by the denominator (or the LCM of the denominators).
When does the inequality sign flip? Only when you multiply or divide by a negative number.
What is the difference between an open and closed circle? Open (hollow) for < and >, where the value is excluded. Closed (filled) for ≤ and ≥, where it is included.
When do I use a dashed line? For strict inequalities (< or >). Solid for ≤ or ≥.
How do I know which side to shade? Substitute a test point such as (0, 0). If it satisfies the inequality, shade its side.
Should my answer be an equation or an inequality? If you were given an inequality, your answer must be an inequality.
Can x be a fraction or negative? Yes, unless the context (lengths, counts of objects) rules it out.
Quick revision checklist
- I do the same operation to both sides, every time
- I can expand brackets correctly before solving
- I can clear fractions by multiplying every term
- I collect letters on the side that keeps them positive
- I check my answer by substituting back
- I can form an equation from words and define my variable
- I answer the question actually asked
- I know all four inequality symbols and their circles
- I flip the sign when multiplying or dividing by a negative
- I know that adding or subtracting does not flip it
- I can solve a double inequality by working on all three parts
- I can list integer solutions, checking both endpoints
- I give an inequality as my answer
- I know dashed vs solid lines for regions
- I use a test point to choose the side to shade
- I check whether the required region should be shaded or left clear, and label it
These notes cover linear equations and inequalities in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.
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