Practice Algebraic Constraint in Math

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

Worksheet PDF Free Answer-key PDF Family

Quick Recap

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

x2+y2=1x^2 + y^2 = 1 constrains (x,y)(x, y) to lie on a circle — not all points in the plane are allowed.

Showing a random 20 of 50 problems.

Example 1

easy
What constraint does x\sqrt{x} place on xx (real numbers)?

Example 2

medium
Why can the constraint x>0x > 0 and x<0x < 0 never be satisfied together?

Example 3

hard
For what real kk does x2+kx+9=0x^2 + kx + 9 = 0 have two distinct real roots?

Example 4

medium
The constraint 2x+3=3+2x2x + 3 = 3 + 2x restricts xx to what set?

Example 5

medium
For x+1x−2\frac{x+1}{x-2}, state all domain constraints.

Example 6

easy
What domain constraint does 1x2−9\dfrac{1}{x^2 - 9} impose?

Example 7

medium
How many integer solutions (x,y)(x,y) satisfy x+y=6x + y = 6 with x,y≥1x, y \ge 1?

Example 8

easy
What constraint on xx makes log⁡(x)\log(x) defined (real, principal branch)?

Example 9

challenge
For x,y>0x, y > 0 with xy=4xy = 4, find the minimum of x+yx + y.

Example 10

medium
Maximize 4x+3y4x + 3y subject to x+y≤4x+y \le 4, x≥0x \ge 0, y≥0y \ge 0.

Example 11

easy
What domain constraint does x−3\sqrt{x-3} impose (over the reals)?

Example 12

medium
If x+y=10x + y = 10 and x,y>0x, y > 0, what is the maximum of xyxy?

Example 13

hard
x+1=x−1\sqrt{x+1} = x - 1. Solve and check.

Example 14

challenge
Find the minimum of x2+y2x^2 + y^2 subject to x+2y=5x + 2y = 5.

Example 15

medium
What values does 1x−3\frac{1}{x-3} constrain xx to avoid?

Example 16

hard
xx−2≥0\dfrac{x}{x-2} \ge 0. Solve.

Example 17

medium
A rectangle has perimeter 2020 with length ℓ\ell and width ww. Write the constraint and the implicit ones.

Example 18

easy
Which describes a constraint: x+3x + 3 or x+3=7x + 3 = 7?

Example 19

challenge
Find all integer points satisfying x+y=4x + y = 4 with x,y≥0x, y \ge 0.

Example 20

medium
Solve x2=9x^2 = 9 subject to the constraint x>0x > 0.
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