Practice Constraints in Math

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

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Quick Recap

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.

Showing a random 20 of 50 problems.

Example 1

easy
A coach allows no more than 4 misses. Write the constraint on misses m.

Example 2

easy
A recipe needs a whole number of eggs and you compute x=2.5. What does the integer constraint tell you?

Example 3

easy
A problem says 0<x<10. Is x=0 allowed?

Example 4

medium
Translate 'no more than 250 but at least 100' to an inequality on x.

Example 5

medium
You have $50 to spend on notebooks ($3 each) and pens ($2 each). Write the constraint inequality and find a valid combination.

Example 6

challenge
Find all integers x with x2<30 and x>0.

Example 7

hard
A recipe needs at least 2 cups of flour and no more than 5 cups total of flour and sugar combined. If you use 3 cups of flour, what is the range of sugar cups?

Example 8

medium
A rectangle has perimeter 20 and integer side lengths. What constraint does each side s satisfy?

Example 9

medium
Determine which values of x∈{−3,0,4,8,12} satisfy ∣x−5∣<4.

Example 10

easy
True or false: 'x>5' includes x=5.

Example 11

easy
A problem says 'the number of apples x must be a whole number ≥0.' Can x=−2?

Example 12

hard
A field has perimeter at most 40 m and each side at least 5 m. Range of width w?

Example 13

easy
List the integer values of x allowed by the constraint 1≤x≤4.

Example 14

challenge
How many ordered pairs of positive integers (x,y) satisfy x+y≤4?

Example 15

hard
A baker uses 1 cup flour per loaf and has 20 cups. Also each loaf takes 30 min and the shift is 5 hr. Max loaves?

Example 16

medium
You have $80 to buy books at $12 each. Write the constraint on number of books n and find the maximum.

Example 17

medium
A box must hold at least 10 items but no more than 25 items. Write this as a compound inequality and list two valid values.

Example 18

hard
Solve x+23≥4 for x.

Example 19

challenge
A pen needs 30 m of fencing along three sides (the fourth is a wall). Each fenced side must be at least 5 m. Find the maximum area.

Example 20

medium
A seating chart needs the number of tables t to satisfy 5t≥38 (each seats 5). What is the smallest valid whole number t?
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