Constraints Formula

The Formula

x + y \leq 100, \quad t \geq 0, \quad x \neq 0

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

Quick Example

Budget: x + y \leq 100 Time: t \geq 0 Domain: x \neq 0

Notation

Constraints are expressed as inequalities (\leq, \geq, <, >) or restrictions (\neq)

What This Formula Means

Conditions or restrictions that limit which values are allowed in a problem. Constraints narrow the set of possible solutions, such as 'x must be positive' or 'the total cannot exceed 100.'

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

Formal View

\text{Feasible set } S = \{x \in D : g_i(x) \leq 0, \; h_j(x) = 0 \; \forall i, j\}

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.

Common Mistakes

  • Solving a problem correctly but ignoring constraints — finding x = -3 when the context requires x > 0
  • Forgetting implicit constraints like 'length must be positive' or 'number of items must be a whole number'
  • Writing \leq when the constraint should be < (strict vs. inclusive inequality)

Why This Formula Matters

Every real-world optimization problem involves constraints — budgets, physical limits, time windows, and rules that define what solutions are actually feasible.

Frequently Asked Questions

What is the Constraints formula?

Conditions or restrictions that limit which values are allowed in a problem. Constraints narrow the set of possible solutions, such as 'x must be positive' or 'the total cannot exceed 100.'

How do you use the Constraints formula?

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

What do the symbols mean in the Constraints formula?

Constraints are expressed as inequalities (\leq, \geq, <, >) or restrictions (\neq)

Why is the Constraints formula important in Math?

Every real-world optimization problem involves constraints — budgets, physical limits, time windows, and rules that define what solutions are actually feasible.

What do students get wrong about Constraints?

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

What should I learn before the Constraints formula?

Before studying the Constraints formula, you should understand: inequalities.

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