Practice Constraints (Meta) in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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?

Showing a random 20 of 50 problems.

Example 1

easy
In 'find a positive integer xx with x<5x < 5', list the values the constraints allow.

Example 2

easy
In log⁡(x)\log(x), what constraint must xx satisfy?

Example 3

challenge
A linear system has constraints x+y≤4x+y \le 4, x≥0x \ge 0, y≥0y \ge 0. Determine the vertices of the feasible region and which constraint is redundant if we add x≤10x \le 10.

Example 4

medium
A factory has 200200 machine-hours and each unit needs 44 hours. Constraint on units produced uu?

Example 5

medium
In an optimization, removing a redundant constraint changes the optimal value how?

Example 6

easy
In ln⁡(x)\ln(x), state the domain constraint.

Example 7

medium
In 9−x2\sqrt{9 - x^2}, give the domain constraint.

Example 8

medium
A constraint says ∣x∣≤3|x| \le 3. Describe the feasible set and contrast it with ∣x∣≥3|x| \ge 3.

Example 9

medium
'Find integers x,y≥0x,y \ge 0 with 2x+3y=122x+3y=12.' How do the constraints reduce the infinitely many real solutions to a finite list?

Example 10

hard
Maximize A=xyA = xy subject to x+y=12x + y = 12, x,y≥0x, y \ge 0.

Example 11

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

Example 12

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

Example 13

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

Example 14

challenge
Find all integer pairs (x,y)(x,y) with x2+y2=25x^2 + y^2 = 25. Explain how the constraint type (sum of two squares = perfect square) limits the search.

Example 15

medium
A rectangle has perimeter 2020 and integer sides. List all (l,w)(l, w) with l≥wl \ge w.

Example 16

hard
A rectangle of perimeter PP has length and width ≥1\ge 1. Constraint on PP?

Example 17

medium
In xx2−4\frac{x}{x^2 - 4}, list excluded values.

Example 18

hard
Find positive integers xx where xx is even AND xx divides 2424.

Example 19

easy
Solving x2=9x^2=9 where the problem says xx is a side length, which root is excluded by the constraint?

Example 20

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