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

Constraints are conditions, rules, or boundaries that restrict which values or solutions are allowed in a mathematical problem, narrowing an infinite space of possibilities to a manageable set.

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 are conditions that shrink an infinite space of possibilities down to the feasible set of values the solution is allowed to take.

Common stuck point: The procedure for constraints (meta) is the easy part; the trap is solving freely and ignoring a stated limit. Asking "Is this a rule that disqualifies otherwise-valid answers, leaving only the ones that satisfy it?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is this a rule that disqualifies otherwise-valid answers, leaving only the ones that satisfy it?

Worked Examples

Example 1

easy

Solve

\frac{1}{x-2} = 3

and identify all constraints on

x

before solving.

Answer

x = \frac{7}{3}

First step

Constraint:

x - 2 \ne 0

, i.e.,

x \ne 2

(denominator cannot be zero).

Full solution

2
Multiply both sides by $(x-2)$ : $1 = 3(x-2)$ .
3
Expand: $1 = 3x - 6$ , so $3x = 7$ , giving $x = \frac{7}{3}$ .
4
Check constraint: $\frac{7}{3} \ne 2$ . Valid.

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

Example 3

medium

Solve

\sqrt{x + 1} = 4

and state the constraint that ensures the solution is valid.

Example 4

medium

A factory has

200

machine-hours and each unit needs

4

hours. Constraint on units produced

u

Example 5

hard

Maximize

A = xy

subject to

x + y = 12

x, y \ge 0

Example 6

challenge

A box has

V = 60

^3

and integer dimensions

l, w, h \ge 1

. How many ordered triples

(l, w, h)

satisfy

V = 60

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

Example 3

easy

In 'find a positive integer

x

with

x < 5

', list the values the constraints allow.

Example 4

easy

'Maximize the area of a rectangle with perimeter

20

.' Which part is the constraint and which is the objective?

Example 5

easy

A recipe requires whole eggs. What implicit constraint does this place on the number of eggs?

Example 6

easy

\log(x)

, what constraint must

x

satisfy?

Example 7

easy

A problem says

x

is a probability. What constraint bounds

x

Example 8

easy

True or false: a constraint tells you what to maximize.

Example 9

easy

Solving

x^2=9

where the problem says

x

is a side length, which root is excluded by the constraint?

Example 10

easy

A scheduling problem requires two meetings not to overlap. Is 'no overlap' a constraint or an objective?

Example 11

medium

A system requires

x+y=10

and

x+y=12

simultaneously. What is the status of this constraint set?

Example 12

medium

In a linear program, why does the optimum often occur at a corner of the feasible region?

Example 13

medium

'Find integers

x,y \ge 0

with

2x+3y=12

.' How do the constraints reduce the infinitely many real solutions to a finite list?

Example 14

medium

A constraint says

|x| \le 3

. Describe the feasible set and contrast it with

|x| \ge 3

Example 15

medium

A factory can make at most

100

units and must make at least

20

. Express the feasible production

p

and the number of integer choices.

Example 16

medium

Why can adding MORE constraints to an optimization never improve the optimal objective value (for a maximization)?

Example 17

medium

A puzzle states '

x

is a digit and

x \ge 5

and

x

is even.' List the feasible digits.

Example 18

medium

In 'cut a

12

-cm wire into two pieces, each at least

3

cm', what constraint governs the cut position

x

(length of first piece)?

Example 19

medium

A problem requires

x

to be a perfect square AND

10 \le x \le 40

. List the feasible values.

Example 20

challenge

Maximize

xy

subject to

x+y=10

with

x,y \ge 0

. Use the constraint to reduce variables and find the maximum.

Example 21

challenge

A linear system has constraints

x+y \le 4

x \ge 0

y \ge 0

. Determine the vertices of the feasible region and which constraint is redundant if we add

x \le 10

Example 22

challenge

Find all integer pairs

(x,y)

with

x^2 + y^2 = 25

. Explain how the constraint type (sum of two squares = perfect square) limits the search.

Example 23

easy

\frac{1}{x-3}

, what value of

x

is excluded by the implicit constraint?

Example 24

easy

True or false: a constraint expands the set of allowed solutions.

Example 25

easy

A triangle has sides

a, b, c

. What constraint must each side satisfy individually?

Example 26

medium

Sides

4, 7, k

form a triangle. What constraint must

k

satisfy?

Example 27

medium

A rectangle has perimeter

20

and integer sides. List all

(l, w)

with

l \ge w

Example 28

medium

\frac{x}{x^2 - 4}

, list excluded values.

Example 29

medium

List integers satisfying

x^2 \le 16

Example 30

medium

In dividing

24

items into groups of equal size, what constraint does the group size satisfy?

Example 31

hard

How many positive divisors of

36

satisfy the constraint 'divisor

\le 12

Example 32

hard

How many integer pairs

(x, y)

satisfy

|x| + |y| \le 2

Example 33

hard

A rectangle of perimeter

P

has length and width

\ge 1

. Constraint on

P

Example 34

hard

Find positive integers

x

where

x

is even AND

x

divides

24

Example 35

challenge

Find all integer

x

with

x^2 - 5x + 6 \le 0

Example 36

challenge

Find all real

x

satisfying both

x^2 < 16

and

x > 0

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Related Concepts

Assumptions Degrees of Freedom

Background Knowledge

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

assumptions