Reasoning with Equations & Inequalities

Before you solve anything, translate. Most of the difficulty in a word problem is turning English into algebra.

"Five less than a number is twelve." Write that as an equation.

Reading it left to right gives 5 - x = 12. That is the natural thing to do, and it is wrong.

"five less than a number is twelve" five less than a number is twelve 5 − x = 12 reads left to right x − 5 = 12 takes 5 away from x ✗ ✓ "less than" tells you what is taken away, not what comes first

Say it aloud instead. Five less than a number — you start with the number, and five is taken off it.

So it is x - 5 = 12, and x = 17. Check: five less than 17 is indeed 12.

The wrong version gives x = -7, which also solves its own equation perfectly. Nothing in the algebra warns you.

This is why translating is the hard part, not solving. English word order and mathematical order are not the same, and "less than", "subtract from" and "fewer than" all reverse it.

A balance scale showing an equation stays level 3x + 5 20 = take 5 from one side, you must take 5 from the other — or it tips

Translate the words

Build the habit of writing what each letter means before anything else. "Let b = the number of blue marbles." One sentence, and half the confusion disappears.

Sum, more than, increased by → +
Difference, less than, decreased by → -
Of, product, times, each → ×
Per, quotient, shared equally → ÷
Is, was, gives, results in → =

"Less than" flips the order

"5 less than x" is x - 5, not 5 - x. Read it as "take 5 away from x". The same flip happens with "subtracted from".

Naming the unknown well

When two quantities are compared, name the smaller or simpler one x and build the other from it.

"Maria has 7 more than twice as many as Jon." Let j = Jon's amount; then Maria has 2j + 7. Choosing Jon keeps the algebra clean.

If instead you set m = Maria's, you would be forced into m - 72 for Jon — correct but messier.

The balance-scale picture

An equation is a balanced scale. Whatever you do to one side, do to the other, or it tips.

That is the entire justification for "subtract 3 from both sides". You are keeping the scale level while you clear away what you do not want.

What inequalities mean

x > 5 does not name one number — it names all of them past 5.

≥ and ≤ include the endpoint; > and < do not. On a number line, a closed dot includes, an open dot excludes.

Phrases to recognise: "at least" means ≥, "at most" means ≤, "more than" means >, "no more than" means ≤.

Does your answer make sense?

Always return to the story. If you find x = -3 chairs, or 2.4 people, something went wrong — or the answer needs rounding in the direction the story requires.

If 4.2 buses are needed, you need 5 buses. Rounding follows the situation, not the usual rule.

Worked examples

Translate and solve

Three less than four times a number is 29. Find the number.

  1. Let n = the number. Name it first, every time.
  2. "Four times a number" is 4n. Build the expression piece by piece.
  3. "Three less than" that is 4n - 3. Take 3 away from 4n — the order flips.
  4. 4n - 3 = 29 ⇒ 4n = 32 ⇒ n = 8 Add 3 to both sides, then divide by 4.

n = 8

Two related unknowns

A rope 84 cm long is cut into two pieces, one three times as long as the other. Find both lengths.

  1. Let s = the shorter piece. Name the simpler quantity.
  2. Longer piece = 3s. Build the second from the first.
  3. s + 3s = 84 ⇒ 4s = 84 The pieces together make the whole rope.
  4. s = 21, so 3s = 63 Check: 21 + 63 = 84 ✓

21 cm and 63 cm.

An inequality from a story

Devon has $50 and buys a $12 ticket. Snacks cost $4 each. How many can he buy?

  1. Let s = number of snacks. Name the unknown.
  2. 12 + 4s ≤ 50 Total spending can be at most $50.
  3. 4s ≤ 38 ⇒ s ≤ 9.5 Subtract 12, then divide by 4.
  4. He cannot buy half a snack. Round down — the story demands it.

At most 9 snacks.

Words to know

Variable
A letter standing for an unknown number.
Equation
A statement that two expressions are equal.
Inequality
A statement that one expression is less than or greater than another.
Solution
A value that makes the statement true.

Mistakes to watch for

Writing "5 less than x" as 5 - x. It is x - 5: the phrase says five is taken off x, so x must come first. "Subtract from" reverses in the same way.

Skipping the "let x = …" step. Without it you can solve correctly and still not know what the answer means — was x the number of adults, or the money they spent? Write it down before the first line.

Rounding by the usual rule instead of by what the situation allows. If 9.4 buses are needed, the answer is 10 — you cannot send four tenths of a bus. Ask what the context permits.

Answering with x when the question wanted something else. Many problems define x as one part and then ask for the total. Re-read the question after solving, every time.

Before symbols, all algebra looked like this. Problems were written as paragraphs and solved in prose — the style is called rhetorical algebra.

A 9th-century problem might read: "a square and ten roots are equal to thirty-nine dirhams." That is x2 + 10x = 39.

Every equation you write is a translation out of a language mathematics spent a thousand years escaping.

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