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 44 misses. Write the constraint on misses mm.

Example 2

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

Example 3

easy
A problem says 0<x<100<x<10. Is x=0x=0 allowed?

Example 4

medium
Translate 'no more than 250250 but at least 100100' to an inequality on xx.

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 xx with x2<30x^2<30 and x>0x>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 2020 and integer side lengths. What constraint does each side ss satisfy?

Example 9

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

Example 10

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

Example 11

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

Example 12

hard
A field has perimeter at most 4040 m and each side at least 55 m. Range of width ww?

Example 13

easy
List the integer values of xx allowed by the constraint 1≤x≤41\le x\le 4.

Example 14

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

Example 15

hard
A baker uses 11 cup flour per loaf and has 2020 cups. Also each loaf takes 3030 min and the shift is 55 hr. Max loaves?

Example 16

medium
You have $80\$80 to buy books at $12\$12 each. Write the constraint on number of books nn 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\frac{x + 2}{3} \ge 4 for xx.

Example 19

challenge
A pen needs 3030 m of fencing along three sides (the fourth is a wall). Each fenced side must be at least 55 m. Find the maximum area.

Example 20

medium
A seating chart needs the number of tables tt to satisfy 5t≥385t\ge 38 (each seats 55). What is the smallest valid whole number tt?
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