Skill: Write equations and inequalities from contexts

Start by identifying:

1. What is changing—use a variable for that quantity.
2. What stays fixed—write it as a number.
3. The relationship—look for words such as total, at most, or at least.
4. The units—make sure each term represents the same kind of quantity.

An equation describes an exact amount. An inequality describes a range of possible amounts.

Example

A game center charges a $3 entry fee plus $2 for each game played. You have at most $11. How many games can you play?

Let $g$ represent the number of games.

The total cost has two parts:

• Entry fee: 3$
• Game cost: 2$per game, or$(2g)$

“At most $11” means the cost can be $11 or less, so use $\leq$:

$$2g+3\leq 11$$

Now solve the inequality step by step:

Subtract $3$ from both sides:

$$2g\leq 8$$

Divide both sides by $2$:

$$g\leq 4$$

Because $g$ is the number of games, it must be a nonnegative whole number. Therefore, you can play

$$0,\ 1,\ 2,\ 3,\ \text{or }4\text{ games}.$$

On a number line, use a closed dot at $4$ because $4$ is included, and shade to the left because all values less than $4$ work:

0----1----2----3----●----5---->
                    4
<===================

The answer in context is: You can play at most 4 games. Checking $(g=4)$, the cost is 11)$, which fits the budget.