Constraints Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Constraints.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Conditions or limitations that restrict which values a variable or solution can take in a problem.

You can't spend more money than you have—that's a constraint.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Constraints define what's possible, limiting the solution space.

Common stuck point: Hidden constraints such as 'number of people must be a whole number' — check the context before finalizing answers.

Sense of Study hint: Write down every restriction the problem gives you before solving, and check your final answer against each one.

Worked Examples

Example 1

medium
You have \50 to spend on notebooks (\3 each) and pens (\$2 each). Write the constraint inequality and find a valid combination.

Solution

  1. 1
    Let \(n\) = notebooks, \(p\) = pens.
  2. 2
    Constraint: \(3n + 2p \leq 50\).
  3. 3
    Also: \(n \geq 0\) and \(p \geq 0\) (non-negativity).
  4. 4
    Valid combo: \(n=10, p=10\): \(3(10)+2(10)=50 \leq 50\) ✓

Answer

Constraint: \(3n + 2p \leq 50\); example: 10 notebooks and 10 pens
A constraint limits the feasible choices. Here the budget constraint \(3n + 2p \leq 50\) defines the region of possible purchases.

Example 2

hard
A farmer plants corn (\200/acre profit) and soybeans (\150/acre profit) on at most 100 acres, with at least 20 acres of corn. Write the constraints and find the profit-maximizing allocation.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
A box must hold at least 10 items but no more than 25 items. Write this as a compound inequality and list two valid values.

Example 2

hard
A recipe needs at least 2 cups of flour and no more than 5 cups total of flour and sugar combined. If you use 3 cups of flour, what is the range of sugar cups?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

inequalities