Constraints Examples in Math

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

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

Concept 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.

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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: A constraint is a condition that fences off which values a variable is permitted to take.

Common stuck point: The procedure for constraints is the easy part; the trap is replacing an inequality constraint with an equation and giving one value. Asking "Does the condition limit or forbid certain values rather than compute one?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the condition limit or forbid certain values rather than compute one?

Worked Examples

Example 1

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

Answer

Constraint: 3n+2p≤503n + 2p \leq 50; example: 10 notebooks and 10 pens

First step

1
Let nn = notebooks, pp = pens.

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Example 2

hard
A farmer plants corn (\$200/acre profit) and soybeans (\$150/acre profit) on at most 100 acres, with at least 20 acres of corn. Write the constraints and find the profit-maximizing allocation.

Example 3

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 4

medium
A ride requires riders to be at least 4040 inches tall AND no more than 8080 inches. Write and use the compound constraint.

Example 5

hard
A box must weigh between 55 kg and 2020 kg, inclusive. Inside are xx books at 0.50.5 kg each. Find the integer range of xx.

Example 6

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.

Practice Problems

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

Example 1

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 2

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 3

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

Example 4

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

Example 5

easy
A solution to an equation is x=−3x=-3, but the problem needs a positive length. Is x=−3x=-3 valid?

Example 6

easy
Write the constraint 'the total cost cannot exceed $100\$100' as an inequality (cost =c=c).

Example 7

easy
Does x=5x=5 satisfy the constraint x>5x>5?

Example 8

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 9

easy
Which values does x≤7x\le 7 allow?

Example 10

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

Example 11

medium
Solve 2x+1=72x+1=7, but the problem requires xx to be a positive even integer. Is the solution acceptable?

Example 12

medium
You have $50\$50 and items cost $8\$8 each. Write and use a constraint for the number nn you can buy.

Example 13

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?

Example 14

medium
A number xx must satisfy both x>0x>0 and x2=9x^2=9. Which solution is valid?

Example 15

medium
A rectangle has perimeter 2020 and integer side lengths. What constraint does each side ss satisfy?

Example 16

medium
Translate: 'at least 33 but fewer than 88' into a single constraint on xx.

Example 17

medium
A number xx satisfies x>4x>4 and x<10x<10. List the integers possible.

Example 18

medium
To ride a coaster you must be at least 4848 inches. Express the constraint and decide if h=47.5h=47.5 qualifies.

Example 19

medium
Two constraints: x≥2x\ge 2 and x≤2x\le 2. What values satisfy both?

Example 20

challenge
Find all integers xx with x2<30x^2<30 and x>0x>0.

Example 21

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

Example 22

challenge
A farmer has 2424 m of fencing for a rectangular pen against a wall (one side needs no fence). The width ww satisfies what constraint, and what is the max area?

Example 23

easy
List integers xx with −2≤x≤3-2 \le x \le 3.

Example 24

easy
A truck holds at most 1,5001{,}500 kg. Write the constraint.

Example 25

easy
A coach allows no more than 44 misses. Write the constraint on misses mm.

Example 26

medium
A baker needs at least 44 cups but at most 99 cups of flour. Write the compound inequality.

Example 27

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 28

medium
Solve 3x−1≤113x - 1 \le 11 and state the constraint on xx.

Example 29

medium
A test has 5050 questions and you must answer at least 3535 correctly to pass. Write the constraint on correct answers cc.

Example 30

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

Example 31

medium
Solve −2x+5>1-2x + 5 > 1 for xx and write the constraint.

Example 32

hard
You need to buy snacks: chips at $2\$2 and drinks at $3\$3, with $24\$24 to spend. Write the constraint and find the maximum total items.

Example 33

hard
How many integers satisfy ∣2x−7∣≤5|2x - 7| \le 5?

Example 34

hard
Solve x+23≥4\frac{x + 2}{3} \ge 4 for xx.

Example 35

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 36

challenge
How many ordered pairs of positive integers (x,y)(x, y) satisfy x+y≤6x + y \le 6?

Example 37

challenge
Find all integers xx with x2≤49x^2 \le 49 and x>−4x > -4.
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Background Knowledge

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

inequalities