Modeling with
Constraints

Learn to represent real-world limits using equations and inequalities, build systems, and interpret which solutions are actually viable.

Algebra 2

What is a Constraint?

A constraint is any condition that limits the possible values of your variables. In the real world, constraints come from budgets, time limits, physical capacity, and rules. In algebra, we express these limits as equations or inequalities.

Equation Constraint

An exact relationship: $ 2x + 3y = 12 $. The values must satisfy the equation precisely.

Inequality Constraint

A range of acceptable values: $ 2x + 3y \leq 12 $. Many solutions work, not just one line.

Why Constraints Matter

Every real decision involves trade-offs. You can't spend more money than you have, you can't work more hours than exist in a day, and a factory can't produce negative items. Constraints capture these truths mathematically so we can find the best possible solution within realistic limits.

The Feasible Region

Translating Words into Math

The key skill is converting verbal descriptions into algebraic expressions. Each resource limit becomes an inequality, and each exact requirement becomes an equation.

1

Identify variables: What quantities are you choosing? Label them $ x, y, z, \ldots $

2

"At most" or "no more than" translates to $ \leq $

3

"At least" or "no fewer than" translates to $ \geq $

4

"Exactly" or "must equal" translates to $ = $

5

Non-negativity: In most real contexts, $ x \geq 0 $ and $ y \geq 0 $

Structure of a Constraint System

Viable Solution

A point $ (x,y) $ that satisfies all constraints and makes sense in context (e.g., whole-number quantities).

Non-viable Solution

A point that violates at least one constraint, or produces an impossible real-world result (e.g., negative items).

Equations vs Inequalities

	Equation	Inequality
Symbol	$ = $	$ \leq,\; \geq,\; <,\; > $
Graph	A line (or curve)	A shaded region
Solutions	Points on the line	Points in the region
Use Case	Exact requirement	Upper/lower bound
Example	$ x + y = 100 $	$ x + y \leq 100 $

How to Build a Constraint Model

1

Read carefully: Identify what you're choosing (the decision variables).

2

Define variables: Let $ x $ = first quantity, $ y $ = second quantity, etc.

3

Write constraints: Translate each limit or requirement into an equation or inequality.

4

Add non-negativity: Include $ x \geq 0,\; y \geq 0 $ unless negatives make sense.

5

Graph and find the feasible region: Shade the overlapping area of all constraints.

6

Interpret solutions: Check whether candidate points are viable in the original context.

Example 1: Nutrition Problem

A meal planner combines Food A and Food B. Each unit of Food A provides 3 g protein and costs $2. Each unit of Food B provides 5 g protein and costs $4. The planner needs at least 30 g of protein and can spend at most $40.

Example 1: Feasible Region

Example 2: Factory Production

A factory makes tables and chairs. Each table requires 4 hours of labor and 3 units of wood. Each chair requires 2 hours and 1 unit of wood. Available: 80 labor hours and 50 units of wood.

Example 3: Concert Tickets

A venue sells VIP tickets ($x$) at $50 and general tickets ($y$) at $20. The venue holds at most 200 people, must earn at least $5000, and policy requires at least 50 general tickets.

Interpreting Viable vs Non-viable

A mathematical solution might satisfy all algebraic constraints but still be non-viable in context. Always check whether the answer makes real-world sense.

Viable

Satisfies every constraint and makes sense: whole-number items, non-negative values, realistic quantities.

Non-viable

Violates at least one constraint, or produces fractional people, negative hours, or impossible results.

Boundary

Points on a constraint line are technically viable (they satisfy $ \leq $ or $ \geq $) but use a resource to its limit.

Common Mistakes

✕ Flipping the Inequality

Writing $ \leq $ when the problem says "at least" (which means $ \geq $). Read the keywords carefully.

✕ Forgetting Non-negativity

Omitting $ x \geq 0 $ and $ y \geq 0 $. Without these, the feasible region extends into negative values that make no real-world sense.

✕ Ignoring Context

Accepting $ x = 7.3 $ tables as a valid answer. If the variable represents whole objects, round and re-check all constraints.

Constraint Modeling Summary

Step	Action	Key Question
1	Define variables	What am I choosing?
2	Write constraints	What limits exist?
3	Graph the system	Where do all constraints overlap?
4	Find feasible region	What is the solution set?
5	Interpret solutions	Does this make real-world sense?

Question 1 of 5

A student has at most 10 hours per week for two part-time jobs. Job A pays $12/hr and Job B pays $15/hr. The student wants to earn at least $120. Which system models these constraints if $ x $ = hours at Job A and $ y $ = hours at Job B?

Question 2 of 5

Given the system $ \begin{cases} 2x + y \leq 20 \\ x + 3y \leq 24 \\ x \geq 0,\; y \geq 0 \end{cases} $, is the point $ (6, 7) $ a viable solution?

Question 3 of 5

A farmer plants corn ($ x $ acres) and wheat ($ y $ acres) on 60 acres. Corn needs 2 workers per acre and wheat needs 3 workers per acre. The farmer has 150 workers. Which pair of constraints represents these limits?

Question 4 of 5

The feasible region of a system of constraints is the set of points that:

Question 5 of 5

A store sells gift baskets with chocolates ($ x $) and candles ($ y $). Each basket must contain at least 5 items and no more than 8 items, with at least 2 chocolates. Which point is a non-viable solution?