Constraints Examples: 43 Problems with Answers

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 \leq 50

; example: 10 notebooks and 10 pens

First step

Let

n

= notebooks,

p

= 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

to buy books at

\$12

each. Write the constraint on number of books

n

and find the maximum.

Example 4

medium

A ride requires riders to be at least

40

inches tall AND no more than

80

inches. Write and use the compound constraint.

Example 5

hard

A box must weigh between

5

kg and

20

kg, inclusive. Inside are

x

books at

0.5

kg each. Find the integer range of

x

Example 6

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.

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

x

must be a whole number

\ge 0

.' Can

x=-2

Example 4

easy

List the integer values of

x

allowed by the constraint

1\le x\le 4

Example 5

easy

A solution to an equation is

x=-3

, but the problem needs a positive length. Is

x=-3

valid?

Example 6

easy

Write the constraint 'the total cost cannot exceed

\$100

' as an inequality (cost

=c

Example 7

easy

Does

x=5

satisfy the constraint

x>5

Example 8

easy

A recipe needs a whole number of eggs and you compute

x=2.5

. What does the integer constraint tell you?

Example 9

easy

Which values does

x\le 7

allow?

Example 10

easy

A problem says

0<x<10

. Is

x=0

allowed?

Example 11

medium

Solve

2x+1=7

, but the problem requires

x

to be a positive even integer. Is the solution acceptable?

Example 12

medium

You have

\$50

and items cost

\$8

each. Write and use a constraint for the number

n

you can buy.

Example 13

medium

A seating chart needs the number of tables

t

to satisfy

5t\ge 38

(each seats

5

). What is the smallest valid whole number

t

Example 14

medium

A number

x

must satisfy both

x>0

and

x^2=9

. Which solution is valid?

Example 15

medium

A rectangle has perimeter

20

and integer side lengths. What constraint does each side

s

satisfy?

Example 16

medium

Translate: 'at least

3

but fewer than

8

' into a single constraint on

x

Example 17

medium

A number

x

satisfies

x>4

and

x<10

. List the integers possible.

Example 18

medium

To ride a coaster you must be at least

48

inches. Express the constraint and decide if

h=47.5

qualifies.

Example 19

medium

Two constraints:

x\ge 2

and

x\le 2

. What values satisfy both?

Example 20

challenge

Find all integers

x

with

x^2<30

and

x>0

Example 21

challenge

How many ordered pairs of positive integers

(x,y)

satisfy

x+y\le 4

Example 22

challenge

A farmer has

24

m of fencing for a rectangular pen against a wall (one side needs no fence). The width

w

satisfies what constraint, and what is the max area?

Example 23

easy

List integers

x

with

-2 \le x \le 3

Example 24

easy

A truck holds at most

1{,}500

kg. Write the constraint.

Example 25

easy

A coach allows no more than

4

misses. Write the constraint on misses

m

Example 26

medium

A baker needs at least

4

cups but at most

9

cups of flour. Write the compound inequality.

Example 27

medium

Determine which values of

x \in \{-3, 0, 4, 8, 12\}

satisfy

|x - 5| < 4

Example 28

medium

Solve

3x - 1 \le 11

and state the constraint on

x

Example 29

medium

A test has

50

questions and you must answer at least

35

correctly to pass. Write the constraint on correct answers

c

Example 30

medium

Translate 'no more than

250

but at least

100

' to an inequality on

x

Example 31

medium

Solve

-2x + 5 > 1

for

x

and write the constraint.

Example 32

hard

You need to buy snacks: chips at

\$2

and drinks at

\$3

, with

\$24

to spend. Write the constraint and find the maximum total items.

Example 33

hard

How many integers satisfy

|2x - 7| \le 5

Example 34

hard

Solve

\frac{x + 2}{3} \ge 4

for

x

Example 35

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 36

challenge

How many ordered pairs of positive integers

(x, y)

satisfy

x + y \le 6

Example 37

challenge

Find all integers

x

with

x^2 \le 49

and

x > -4

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Related Concepts

Inequalities Optimization Systems of Equations

Background Knowledge

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

inequalities