Bounds Examples in Math

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

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

Concept Recap

The upper and lower limits within which a quantity must lie; often expressed as $a \leq x \leq b$ .

Temperature tomorrow will be between 60F and 75F. Those are bounds.

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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: Bounds give the lowest and highest values a quantity is allowed to take, often $a\le x\le b$ .

Common stuck point: The procedure for bounds is the easy part; the trap is stating only one limit. Asking "Is the value pinned by both a smallest and a largest allowed amount?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the value pinned by both a smallest and a largest allowed amount?

Worked Examples

Example 1

medium

For

f(x) = -x^2 + 6x - 5

, find the maximum value (upper bound) on

[0, 5]

Answer

Maximum value = 4 at

x = 3

First step

Find the vertex:

x = -b/(2a) = -6/(2 \times -1) = 3

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Example 2

hard

Show that for all real

x

x^2 \geq 0

. What is the greatest lower bound (infimum) of

x^2

Example 3

medium

For

f(x)=x^2

[-3,2]

, find the upper and lower bounds of

f

Example 4

medium

For

f(x)=\frac{1}{x}

[2,5]

, find the bounds of

f

Example 5

hard

For

f(x)=x^3-3x

[-2,2]

, find the upper and lower bounds.

Example 6

challenge

By AM-GM, find the minimum of

x+\frac{4}{x}

for

x>0

and state the lower bound.

Practice Problems

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

Example 1

medium

For

g(x) = 3x + 1

[0, 4]

, find the minimum and maximum values.

Example 2

hard

Prove that

\sin(x) \leq 1

and

\sin(x) \geq -1

for all

x

using the unit circle definition.

Example 3

easy

A quantity satisfies

3\le x\le 9

. What is its lower bound?

Example 4

easy

Tomorrow's temperature will be between

60^\circ

and

75^\circ

. What is the upper bound?

Example 5

easy

Write 'x is at most

12

and at least

4

' as a bounded inequality.

Example 6

easy

Does the bound

x\le 7

limit how small

x

can be?

Example 7

easy

Between which two integers is

\sqrt{20}

bounded?

Example 8

easy

The interval

2<x<6

uses what kind of bounds, strict or inclusive?

Example 9

easy

Give an upper bound for the value

x

x+5\le 12

Example 10

easy

What is the smallest integer satisfying

x\ge 3.2

Example 11

medium

4\le x\le 9

and

2\le y\le 5

, find the bounds on

x+y

Example 12

medium

2\le x\le 6

, find the bounds on

x-y

where

1\le y\le 4

Example 13

medium

A rounded measurement reads

7

cm to the nearest cm. What are the bounds on the true length

L

Example 14

medium

2\le x\le 6

and

1\le y\le 3

, find the bounds on the product

xy

(all positive).

Example 15

medium

1\le x\le 5

, find the bounds on

10-x

Example 16

medium

3\le x\le 5

, find the bounds on

x^2

Example 17

medium

Find bounds on

2x+3

given

-1\le x\le 4

Example 18

medium

-2\le x\le 3

, find the bounds on

-x

Example 19

medium

A right triangle has legs each between

3

and

4

. Bound its hypotenuse.

Example 20

challenge

Integers

x,y

satisfy

1\le x\le 5

and

x<y\le 6

. How many ordered pairs

(x,y)

exist?

Example 21

challenge

2\le x\le 8

, find the tightest bounds on

\frac{12}{x}

Example 22

challenge

The perimeter of a rectangle is

20

. Bound its area

A

Example 23

easy

State the lower and upper bounds described by

-3 \leq x \leq 8

Example 24

easy

x=5

allowed by the bound

x<5

Example 25

easy

Give an integer that satisfies both

x>4

and

x\le 7

Example 26

easy

Express 'no more than

50

' as a bound on

x

Example 27

medium

-2\le x\le 5

and

1\le y\le 4

, find the tightest bounds on

x+y

Example 28

medium

2\le a\le 5

and

-3\le b\le 1

, find bounds on

a-b

Example 29

medium

A length is measured as

12

cm to the nearest cm. State bounds on the true length

L

Example 30

medium

1\le x\le 4

and

2\le y\le 5

with both positive, find bounds on

x/y

Example 31

medium

Solve

-2x+5\ge 1

and give bounds on

x

Example 32

medium

For

f(x)=\sin x + 2

, find the upper and lower bounds of

f

Example 33

medium

|x-4|\le 3

, write the bounds on

x

Example 34

hard

-3\le x\le 2

, find the tightest bounds on

x^2

Example 35

hard

-2\le x\le 3

and

-4\le y\le 1

, find the tightest bounds on

xy

Example 36

hard

A rectangle has width measured as

4

cm and length

9

cm, each to the nearest cm. Find bounds on the area.

Example 37

hard

2<x<6

and

3<y<5

, find the tightest open bounds on

x+y

Example 38

challenge

Find the tightest bounds on

f(x,y)=\frac{x+y}{x-y}

given

4\le x\le 6

and

1\le y\le 2

Example 39

challenge

For all real

x

, prove

x^2-4x+7\ge 3

and state the lower bound.

Example 40

challenge

A box has

L\in[10,12]

W\in[5,6]

H\in[3,4]

(cm). Find tightest bounds on volume.

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Related Concepts

Inequality Intuition Interval Notation Limit

Background Knowledge

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

inequality intuition