Writing Equations and Inequalities from Words

What solving actually asks

An equation is a claim that two quantities are equal. Solving it means finding which values of the variable make the claim true.

$$3n + 5 = 20$$

That asks: which $n$ makes $3n + 5$ come to $20$?

$n$	$3n + 5$	True?
$3$	$14$	no
$4$	$17$	no
$5$	$20$	yes
$6$	$23$	no

So $n = 5$ is the solution. Testing values is a slow way to find it, and it does show exactly what the answer means: the one value that makes the statement true.

An inequality asks a wider question:

$$3n + 5 > 20$$

Which values of $n$ make the left side exceed $20$? Here every $n$ above $5$ works, so the answer is a range rather than a single number.

Turning words into an equation

Three steps, in order.

1. Name the unknown. Write down what the letter stands for.
2. Find the relationship. What does the problem say is equal to what?
3. Write it down.

A taxi charges $5$ plus $3$ a mile. A ride cost $26$. How far was it?

$$n = \text{number of miles}$$

The cost is $3n + 5$, and that cost was $26$:

$$3n + 5 = 26$$

Naming the unknown is not a formality. It fixes what the answer will mean — here, miles rather than dollars — and a solution of $7$ is meaningless until you know which.

The words that signal each sign

Phrase	Sign	Includes the number?
more than, over, exceeds	$>$	no
less than, under, below	$<$	no
at least, minimum, no less than	$\ge$	yes
at most, maximum, no more than	$\le$	yes

The right-hand column is what most inequality errors come down to.

“You must be at least $18$ to vote” includes someone who is exactly $18$:

$$a \ge 18$$

“A lift holds no more than $8$ people” includes exactly $8$:

$$p \le 8$$

Writing $a > 18$ would exclude every eighteen-year-old, which is not what the rule says.

Two quantities that change together

Some problems have no single unknown. Instead one quantity depends on another, and the equation records the relationship.

A printer prints $9$ pages a minute. Write an equation for pages after $t$ minutes.

$$p = 9t$$

Here $t$ is the independent variable — the one you choose — and $p$ is the dependent one, because its value follows from $t$.

$t$	$p$
$1$	$9$
$5$	$45$
$10$	$90$

This is the same proportional relationship seen from the algebra side, with $9$ as the constant of proportionality.

Why writing the equation is the hard part

Once an equation exists, solving it is mechanical — a fixed procedure that always terminates. Getting to the equation is where the thinking happens, and where most marks are lost.

Two habits do most of the work.

Check the units. In $3n + 5 = 26$ every term is dollars: $3n$ is dollars per mile times miles, $5$ is a flat charge, and $26$ is the fare. If one term came out in miles, the equation would be wrong before any solving started.

Test the equation before solving it. Put in a rough value and see whether the statement behaves sensibly. A $10$-mile ride should cost $35$ by this equation, which is plausible for a taxi. If it produced $305$, the equation is wrong and solving it carefully would only produce a careful wrong answer.

Worked examples

:::example{title=“Example 4 — An inequality with “at least""} A club needs at least $25$ members to run. Write an inequality.

“At least” includes $25$ itself:

$$m \ge 25$$

:::

Common mistakes

Practice problems