One-page revision sheet

• In algebra a letter stands for an unknown number.
• An expression has no equals sign; an equation does.
• Write the number before the letter: x × 3 is written 3x, never x3.
• Never write the 1: 1 × a is written a.
• Write division as a fraction: n ÷ 4 is written n/4.
• a × a is written a² and a × a × a is written a³.
• “More than” means add; “less than” means subtract, but watch the order — “x less than y” is y − x, not x − y.
• “Doubled” means × 2, “tripled” means × 3, “half of” means ÷ 2.

Key words: expression  ·  equation  ·  term  ·  variable  ·  unknown  ·  coefficient  ·  constant  ·  equivalent expression

• A formula is a rule connecting quantities. It always has an equals sign.
• To substitute, replace each letter with its number and then calculate.
• Keep the order of operations: brackets and indices before × ÷, and those before + −.
• Be careful with negatives: when x = −4, 3x² means 3 × (−4)² = 3 × 16 = 48.
• You can derive your own formula. Choose letters that suggest what they stand for, and always say what each one means.
• You can work backwards from a formula: if T = 25h + 40 and T = 190, then h = 6.

Key words: formula  ·  formulae  ·  substitute  ·  derive  ·  variable

• Like terms contain exactly the same letters to the same powers.
• Only like terms can be combined: 3x + 2x = 5x, but 3x + 4y cannot be simplified.
• x and x² are not like terms.
• Deal with each letter separately, and treat the plain numbers as another group.
• Write algebra tidily: a not 1a, 2a not a + a, ab not b × a, a² not a × a.
• A negative coefficient is fine: 4a − 7a = −3a.
• For fraction terms, use a common denominator: 3a/4 + 5a/12 = 9a/12 + 5a/12 = 14a/12 = 7a/6.

Key words: like terms  ·  collecting like terms  ·  simplify  ·  term

• To expand, multiply every term inside the brackets by the term outside.
• 4(n + 3) means 4 × (n + 3), so it equals 4 × n + 4 × 3 = 4n + 12.
• Keep the signs: 2(x − 5) = 2x − 10, not 2x + 10.
• A letter can be outside the brackets too: x(x + 6) = x² + 6x.
• A negative outside changes every sign inside: −2(x + 3) = −2x − 6.
• A minus in front of a bracket means multiply by −1: −(x + 7) = −x − 7.
• After expanding two brackets, collect the like terms.
• Check your expansion by substituting a number — both forms must give the same value.

Key words: brackets  ·  expand  ·  term

• An equation is a balance: the two sides are equal.
• To solve, do the same thing to both sides until the letter is alone.
• Use inverse operations: undo + with −, undo × with ÷.
• Undo in reverse order — deal with the + or − first, then the × or ÷.
• x + 5 = 12 → subtract 5 → x = 7.   2y + 4 = 16 → subtract 4 → 2y = 12 → divide by 2 → y = 6.
• With brackets you can divide by the number outside first: 4(x − 2) = 20 → x − 2 = 5 → x = 7.
• Always check by substituting your solution back into the original equation.

Key words: equation  ·  solve  ·  solution  ·  inverse operation  ·  balance

• < means “is less than”, > means “is greater than”. The wide end always faces the larger quantity.
• x > 5 means x is bigger than 5. The smallest integer is 6, because 5 itself is not allowed.
• y < 2 means y is smaller than 2. The largest integer is 1.
• On a number line use an open circle at the boundary, because that value is not included.
• The arrow points right for > and left for <.
• You cannot list every solution — there are infinitely many — so list the first few and write “…”.
• Two inequalities together trap a value: x > 2 and x < 7 gives the integers 3, 4, 5, 6.
• You solve an inequality just like an equation: 3x + 1 > 16 → 3x > 15 → x > 5.

Key words: inequality  ·  less than (<)  ·  greater than (>)  ·  integer  ·  open circle