Skill: Model contextual problems with equations and inequalities

A contextual problem can be modeled by:

1. Choosing a variable for the unknown quantity.
2. Writing an expression for the total or changing amount.
3. Using an equation for an exact amount.
4. Using an inequality for a limit or range.
5. Checking that the answer makes sense in context.

Example

A taxi charges a fixed fee of $4 plus $2.50 for each mile traveled.

Step 1: Define the variable

Let

$$m=\text{number of miles traveled}.$$

Since distance cannot be negative,

$$m\ge 0.$$

Step 2: Write an expression for the cost

The fixed fee is $4. The mileage charge is $2.50 for each of $m$ miles:

$$\text{cost}=4+2.50m.$$

The units are dollars, because both terms represent dollar amounts.

Step 3: Model an exact condition with an equation

Suppose the ride costs exactly $19. “Exactly” means use an equation:

$$4+2.50m=19.$$

Solve:

$$2.50m=15$$ $$m=6.$$

So, a ride of exactly $6$ miles costs $19. Check:

$$4+2.50(6)=4+15=19.$$

Step 4: Model a limit with an inequality

Suppose instead that a passenger has at most $19 to spend. “At most” means the cost can be $19 or less:

$$4+2.50m\le 19.$$

Solve:

$$2.50m\le 15$$ $$m\le 6.$$

Including the domain restriction $(m\ge 0)$, the possible distances are

$$0\le m\le 6.$$

On a number line, this is shown with a closed dot at $0$, a closed dot at $6$, and shading between them:

$$[0,6]$$

Therefore, the passenger can travel any distance from $0$ to $6$ miles, inclusive. An equation gives the exact distance, while an inequality gives all distances that satisfy the budget limit.