Expressions, equations, and inequalities are mathematical models. They are mathematical representations used to describe quantities and their relationships in a real-life situation. Often, what we want to describe are constraints. A constraint is something that limits what is possible or what is reasonable in a situation.

For example, when planning a birthday party, we might be dealing with these quantities and constraints:

We can use both numbers and letters to represent the quantities. For example, we can write 42 to represent the cost of entertainment, but we might use the letter to represent the number of people at the party and the letter for the total cost in dollars.

We can also write expressions using these numbers and letters. For instance, the expression is a concise way to express the overall cost of food if it costs \$5.50 per guest and there are guests.

Sometimes a constraint is an exact value. For instance, the cost of music is \$15. Other times, a constraint is a boundary or a limit. For instance, the total cost must be no more than \$180. Symbols such as <, >, and = can help us express these constraints.

Equations can show the relationship between different quantities and constraints. For example, the total cost of the party is the sum of the costs of food, cake, and entertainment. We can represent this relationship with:

Deciding how to use numbers and letters to represent quantities, relationships, and constraints is an important part of mathematical modeling. Making assumptions—about the cost of food per person, for example—is also important in modeling.

A model such as can be an efficient way to make estimates or predictions. When a quantity or a constraint changes, or when we want to know something else, we can adjust the model and perform a simple calculation, instead of repeating a series of calculations.