Skill: Write equations and inequalities from contexts

To write an equation or inequality from a situation:

1. Identify the unknown quantity.
2. Choose a variable to represent it.
3. Translate the relationships in the situation into mathematical operations.
4. Use:
   • $=$ when two amounts must be exactly equal.
   • $<$, $>$, $\le$, or $\ge$ when the situation gives a limit or range.
5. Check the meaning and units of your statement.

Example

A movie ticket costs $8. You have $40 to spend. How many tickets can you buy?

Step 1: Choose a variable.

Let $t$ represent the number of movie tickets.

Step 2: Write the total cost.

Each ticket costs $8, so the cost of $t$ tickets is

$$8t \text{ dollars}.$$

Step 3: Identify the limit.

You can spend at most $40. “At most” means the amount can be $40 or less, so use $\le$.

$$8t \le 40$$

This inequality means that the cost of the tickets must not be greater than $40.

Step 4: Check the context.

Since $t$ represents a number of tickets, $t$ should be a nonnegative whole number. In fact, solving the inequality gives

$$t \le 5,$$

so you can buy at most $5$ tickets. The inequality matches the situation because buying fewer than $5$ tickets is also allowed.