Write equations and inequalities from contexts

A learner translates a situation into a one-variable linear equation or inequality by identifying quantities, choosing a variable, and expressing relationships such as totals, rates, comparisons, and constraints with appropriate operations and units. The learner understands that an equation represents a condition requiring equality, whereas an inequality represents a range of values, and checks that the symbolic statement matches the context; systems, nonlinear equations, and more abstract parameterized models are beyond this scope.

Detailed Explanation: Write equations and inequalities from contexts

To write an equation or inequality from a situation:

Identify the unknown quantity.
Choose a variable to represent it.
Translate the relationships in the situation into mathematical operations.
Use:
- $=$ when two amounts must be exactly equal.
- $<$ , $>$ , $\le$ , or $\ge$ when the situation gives a limit or range.
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.

Learn by doing: Write equations and inequalities from contexts

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