Understanding Algebra
This guide explains the why behind every rule — not just what to memorise, but how to think algebraically. Work through each section in order, study each worked example step by step, then test yourself with the checkpoint questions and reveal the answers to check your thinking.
Before algebra, every problem had to be solved with specific numbers. Algebra gave mathematicians a tool to describe any situation — by using a letter to stand for a number we don't know yet, or a number that can change.
Consider the question "what number, when you double it and add 3, gives 11?" You could guess and check — or you could write 2x + 3 = 11 and solve it in two steps. Variables let us generalise: the rule Perimeter = 4 × side works for every square, not just one with a specific side length. That power to generalise is the entire point of algebra.
Key Vocabulary
| Term | Meaning | Example |
|---|---|---|
| Variable | A letter representing an unknown or changing number | x, y, n |
| Term | A single number, variable, or product of both | 5x, −3, 7xy |
| Coefficient | The number multiplying a variable in a term | In 5x, the coefficient is 5 |
| Like terms | Terms with the same variable(s) raised to the same exponents | 3x and −7x are like terms |
| Expression | A combination of terms — no equals sign | 3x + 2y − 5 |
| Equation | Two expressions linked by an equals sign | 3x + 5 = 14 |
Multiply the factor outside the brackets by every term inside. This is the single most-used rule in algebra.
The factor outside is 3. Inside the brackets: +2x and −5.
Be careful: a negative times a positive gives a negative; a negative times a negative gives a positive.
The minus sign in front of 2(3x − 1) means the entire second bracket is being subtracted.
constants: 12 + 2 = 14
Result: −2x + 14
b) Expand: 5(3x − 2)
c) Is 4x + 9 an expression or an equation? What about 4x + 9 = 0?
a) 3 terms. Coefficient of x is 7. Coefficient of y is −3 (the minus sign belongs to the term).
b) 5(3x − 2) = 5 × 3x + 5 × (−2) = 15x − 10
c) 4x + 9 is an expression — no equals sign. 4x + 9 = 0 is an equation — it has an equals sign making a statement about x.
Simplifying means writing an expression in its shortest, clearest form. The two main tools are: combining like terms, and expanding brackets with the distributive property.
Think of terms as physical objects. 3 apples + 2 apples = 5 apples. But 3 apples + 2 oranges cannot be simplified — they are different things. In algebra, 3x and 2x are like terms (both "groups of x") and combine to 5x. But 3x and 2y are different variables — you cannot add them into a single term.
−3(x − 4) = −3x + 12, NOT −3x − 12.
Remember: negative × negative = positive. The −3 multiplies both the x and the −4.
y terms: 3y − y = 2y
constants: 7
constants: 2 + 8 = 10
Result: 2x + 10
b) Expand and simplify: 3(2x − 5) + 4(x + 1)
c) A student writes −5(x − 3) = −5x − 15. Identify the error and give the correct answer.
a) Group: (4a + 2a) + (−3b + b) + (7 − 4) = 6a − 2b + 3
b) 3(2x − 5) = 6x − 15; 4(x + 1) = 4x + 4. Together: 6x − 15 + 4x + 4 = 10x − 11
c) Error: −5 × −3 should be +15, not −15. Correct answer: −5x + 15. (Negative times negative is positive.)
An equation is like a perfectly balanced scale. Both sides weigh the same because they are equal. Your job is to find the value of the unknown variable — and you do that by keeping the scale balanced at every step.
If you have a balanced scale and you add 5 kg to one side, you must add 5 kg to the other side to keep it balanced. Same in algebra: whatever operation you apply to one side of the equation, you must apply the exact same operation to the other side. This is the golden rule that makes algebra work.
Solving Procedure
- Expand any brackets using the distributive property
- Collect all variable terms on one side (add or subtract)
- Collect all constant terms on the other side
- Divide both sides by the coefficient of the variable
- Check your answer by substituting it back into the original equation
3x = 9
x = 3
x − 6 = 7
x = 13
Right: 2(13) + 7 = 26 + 7 = 33 ✓
Denominators are 3 and 2. LCD = 6. Multiply every term on both sides by 6.
2x + 6 = 3x − 12
18 = x
Right: 18/2 − 2 = 9 − 2 = 7 ✓
When you move a term across the equals sign it changes sign. This is really just adding the opposite to both sides. For example: x − 5 = 3 → add 5 to both sides → x = 3 + 5 = 8. Students often write x = 3 − 5 — a sign flip error. Always think "what did I add/subtract to both sides?" rather than mechanically moving a term.
b) Solve and check: 4(x + 2) = 3x + 11
c) Solve: x/4 + 3 = x/6 + 5 (Hint: LCD = 12)
a) 5x = 22 + 3 = 25 → x = 25 ÷ 5 = 5. Check: 5(5) − 3 = 22 ✓
b) Expand: 4x + 8 = 3x + 11 → 4x − 3x = 11 − 8 → x = 3. Check: 4(3+2) = 20 and 3(3)+11 = 20 ✓
c) Multiply by 12: 3x + 36 = 2x + 60 → x = 60 − 36 = 24. Check: 24/4 + 3 = 9 and 24/6 + 5 = 9 ✓
An inequality is a statement that one expression is greater than, less than, or equal to another. You solve inequalities using exactly the same steps as equations — with one critical extra rule.
Picture the number line. We know 3 < 8. Now multiply both sides by −1: −3 and −8. On the number line, −3 is to the right of −8, so −3 > −8. The order reversed! Multiplying by a negative always reflects numbers across zero, swapping left and right — and therefore flipping the inequality direction.
Inequality Symbols
| Symbol | Meaning | Number line |
|---|---|---|
| < | Strictly less than — the left side is smaller | Open circle, arrow pointing left |
| > | Strictly greater than — the left side is larger | Open circle, arrow pointing right |
| ≤ | Less than or equal to — includes the boundary value | Closed (filled) circle, arrow left |
| ≥ | Greater than or equal to — includes the boundary value | Closed (filled) circle, arrow right |
When you multiply or divide both sides of an inequality by a negative number, the inequality sign must be flipped.
Example: −2x < 8 → divide both sides by −2 → x > −4 (sign flipped from < to >).
2x < 8
x < 4
The solution is all real numbers less than 4. On a number line: open circle at 4, arrow pointing left.
−3x ≥ 9
Dividing by a negative number: the ≥ becomes ≤.
x ≤ −3
Test x = 0: 5 − 0 = 5 ≥ 14? No ✗ (correctly excluded)
b) Solve: −2x + 6 ≤ 12 (watch the flip!)
c) A student solves −4x > 20 and writes x > −5. What did they do wrong? Give the correct answer.
a) 4x > 11 + 1 = 12 → x > 12 ÷ 4 = 3. (Divided by positive 4 — no flip.)
b) −2x ≤ 12 − 6 = 6 → divide by −2, flip sign: x ≥ −3.
c) The student divided by −4 to get −5, but forgot to flip the sign. Correct: x < −5. When you divide both sides by −4, the > must become <.
Word problems are where algebra meets the real world. The skill is not in the arithmetic — it is in the translation: reading a sentence and writing it as a mathematical equation.
Every sentence in a word problem contains mathematical clues. "Increased by" means addition. "Times" means multiplication. "Is" means equals. Once you learn to read these clues, translating a paragraph into an equation becomes straightforward — and then you already know how to solve the equation.
1. Read carefully — identify what is unknown and what information is given.
2. Define the variable — write "let x = ..." clearly.
3. Write an equation — translate the key sentence into algebra.
4. Solve the equation — use the steps from Section 3.
5. Answer in a full sentence — state what x represents, include units if needed.
Common Translations
| English phrase | Algebra |
|---|---|
| A number increased by 5 | x + 5 |
| Three times a number | 3x |
| A number decreased by 7 | x − 7 |
| Twice a number minus 4 | 2x − 4 |
| Half of a number | x/2 |
| The sum of two consecutive integers | x + (x + 1) |
−2x = −12
x = 6
Right: 3(6) − 4 = 18 − 4 = 14 ✓
Consecutive integers follow each other in order: 4, 5, 6 or −1, 0, 1, etc. If the first is x, the next is x+1, then x+2.
Then x+1 = second integer
And x+2 = third integer
3x = 54
x = 18
Check: 18 + 19 + 20 = 57 ✓
b) The sum of two consecutive even integers is 46. Find the integers. (Hint: consecutive even integers differ by 2, so let them be x and x+2.)
c) Esmeralda is 3 years younger than her brother Emil. In 5 years, the sum of their ages will be 39. How old is Esmeralda now?
a) Let x = the number. Equation: 2x + 5 = 23 → 2x = 18 → x = 9. Check: 2(9)+5 = 23 ✓
b) Let x = first even integer. x + (x+2) = 46 → 2x + 2 = 46 → 2x = 44 → x = 22. The integers are 22 and 24. Check: 22 + 24 = 46 ✓
c) Let x = Esmeralda's current age, then Emil's age now = x + 3. In 5 years: (x+5) + (x+3+5) = 39 → 2x + 13 = 39 → 2x = 26 → x = 13. Esmeralda is currently 13 years old. Check: Emil is 16; in 5 years they are 18 and 21; 18 + 21 = 39 ✓