Algebraic Constraint Examples in Math

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

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

Concept Recap

A mathematical condition expressed as an equation or inequality that restricts which values the variables are allowed to take.

$x^2 + y^2 = 1$ constrains $(x, y)$ to lie on a circle — not all points in the plane are allowed.

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: An algebraic constraint is an equation or inequality saying which values are allowed.

Common stuck point: The procedure for algebraic constraint is the easy part; the trap is ignoring implicit constraints like denominator $\ne0$ or radicand $\ge0$ . Asking "Does this condition restrict the set of permissible values rather than ask for a single computation?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does this condition restrict the set of permissible values rather than ask for a single computation?

Worked Examples

Example 1

easy

What constraint does

x^2 \geq 0

impose on

x

Answer

No restriction —

x^2 \geq 0

is always true for real

x

First step

Step 1:

x^2 \geq 0

for all real

x

— this is always true.

Full solution

2
Step 2: This constraint doesn't restrict $x$ at all; every real number satisfies it.
3
Step 3: But $x^2 = -1$ would impose an impossible constraint (no real solution).

Some constraints are automatically satisfied (trivial constraints), while others like

x^2 = -1

are impossible. Understanding what a constraint rules out is as important as what it allows.

Example 2

medium

What values does

\frac{1}{x-3}

constrain

x

to avoid?

Example 3

medium

A box has length

\ell

, width

w

, and height

h

, all positive, with total surface area

\le 96

. Write all constraints.

Example 4

medium

(x-2)(x+3) \ge 0

. Solve.

Example 5

hard

A budget of \$60 buys apples at \$3 each and bananas at \$2 each, with at least 5 apples. Find the maximum total fruit.

Example 6

challenge

Solve the system

x + y + z = 6

x, y, z \ge 0

integers. How many solutions?

Practice Problems

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

Example 1

easy

What constraint does

\sqrt{x}

impose on

x

Example 2

medium

x + y = 10

and

x, y > 0

, what is the maximum of

xy

Example 3

easy

What values of

x

are allowed by the constraint

x \ge 0

Example 4

easy

\frac{1}{x}

, what value is forbidden by an implicit constraint?

Example 5

easy

What constraint does

\sqrt{x}

place on

x

(real numbers)?

Example 6

easy

Does the point

(2, 0)

satisfy the constraint

x + y = 2

Example 7

easy

A length

\ell

satisfies what real-world constraint?

Example 8

easy

Which describes a constraint:

x + 3

x + 3 = 7

Example 9

easy

How many values of

x

satisfy the constraint

x = 5

Example 10

easy

Geometrically, what does the constraint

x^2 + y^2 = 1

describe?

Example 11

medium

Combine the constraints

x > 0

and

x < 5

. What is the feasible set?

Example 12

medium

For

\frac{x+1}{x-2}

, state all domain constraints.

Example 13

medium

Solve

x^2 = 9

subject to the constraint

x > 0

Example 14

medium

A rectangle has perimeter

20

with length

\ell

and width

w

. Write the constraint and the implicit ones.

Example 15

medium

(3, 4)

feasible under both

x + y \le 10

and

x \ge 0

Example 16

medium

The constraint

|x| \le 3

describes which values?

Example 17

medium

Why can the constraint

x > 0

and

x < 0

never be satisfied together?

Example 18

medium

The constraint

2x + 3 = 3 + 2x

restricts

x

to what set?

Example 19

medium

Given

x \ge 2

and

x \ge 5

, which constraint is binding?

Example 20

challenge

Maximize

x + y

subject to

x + 2y \le 8

x \ge 0

y \ge 0

. Find the maximum.

Example 21

challenge

Find all integer points satisfying

x + y = 4

with

x, y \ge 0

Example 22

challenge

For what

k

does the constraint

x^2 + y^2 = k

have any real solutions?

Example 23

easy

What is the domain constraint of

\dfrac{1}{x+4}

Example 24

easy

Solve

3x = 12

subject to

x > 0

Example 25

easy

Which point satisfies

x + y \le 5

and

x \ge 0

(2, 4)

(-1, 3)

Example 26

easy

What constraint on

x

makes

\log(x)

defined (real, principal branch)?

Example 27

easy

What domain constraint does

\dfrac{1}{x^2 - 9}

impose?

Example 28

medium

Find the integer values of

x

satisfying

-2 \le x < 3

Example 29

medium

x^2 = 16

subject to

x

negative. Find

x

Example 30

medium

Solve

x^2 = 25

subject to 'side length of a square'. Which root applies?

Example 31

medium

Domain of

f(x) = \dfrac{\sqrt{x-1}}{x-4}

Example 32

medium

How many integer solutions

(x,y)

satisfy

x + y = 6

with

x, y \ge 1

Example 33

medium

A right triangle has legs

a

and

b = a+1

with hypotenuse 5. State the constraint and solve.

Example 34

medium

Maximize

4x + 3y

subject to

x+y \le 4

x \ge 0

y \ge 0

Example 35

hard

\sqrt{x+1} = x - 1

. Solve and check.

Example 36

hard

How many lattice points

(x,y)

satisfy

x \ge 0

y \ge 0

x + 2y \le 6

Example 37

hard

For what real

k

does

x^2 + kx + 9 = 0

have two distinct real roots?

Example 38

hard

\dfrac{x}{x-2} \ge 0

. Solve.

Example 39

hard

For which real

a

is the system

x + y = a

x^2 + y^2 = 1

feasible?

Example 40

challenge

Find the minimum of

x^2 + y^2

subject to

x + 2y = 5

Example 41

challenge

For

x, y > 0

with

xy = 4

, find the minimum of

x + y

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Related Concepts

Equations Inequalities Constraint System Optimization

Background Knowledge

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

equationsinequalities