Constraints (Meta) Examples in Math

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

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 restrictions that bound the set of allowable values or solutions in a problem.

The rules of the game. What must be true? What can't happen?

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 the problem. Often, the constraints ARE the problem.

Common stuck point: Missing a constraint leads to 'solutions' that don't actually work.

Sense of Study hint: List every condition the solution must satisfy, including hidden ones like 'must be positive' or 'must be an integer.' Then test your answer against each.

Worked Examples

Example 1

easy
Solve \frac{1}{x-2} = 3 and identify all constraints on x before solving.

Solution

  1. 1
    Constraint: x - 2 \ne 0, i.e., x \ne 2 (denominator cannot be zero).
  2. 2
    Multiply both sides by (x-2): 1 = 3(x-2).
  3. 3
    Expand: 1 = 3x - 6, so 3x = 7, giving x = \frac{7}{3}.
  4. 4
    Check constraint: \frac{7}{3} \ne 2. Valid.

Answer

x = \frac{7}{3}
Constraints limit the set of allowable values. For rational expressions, the denominator must be non-zero. Checking the solution against constraints is always required.

Example 2

medium
An integer n satisfies two constraints: n > 0 and n < 10, and also n is prime. List all valid values of n.

Practice Problems

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

Example 1

easy
State the domain constraint for f(x) = \sqrt{x - 4} and solve for x when f(x) = 3.

Example 2

medium
A rectangle has area 36 cm² and integer side lengths. List all constraints and find all valid dimension pairs (l, w) with l \ge w.
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Background Knowledge

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

assumptions