ORCCA Comparison Symbols and Notation for Intervals

Section 1.3 Comparison Symbols and Notation for Intervals

Objectives: PCC Course Content and Outcome Guide

MTH 60 CCOG 2.4

Here is a true fact:

8

is larger than

3 .

That is a comparison between two specific numbers. We can also make comparisons using an unspecified number, like if we say that average rent for an apartment in Portland, OR is more than $1700. We are not saying what the average rent is, just that it’s larger than $1700. In the first half of this section, we examine mathematical notation for making these kinds of comparisons.

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In Oregon, only citizens

18

and older can vote in statewide elections. That is saying something about a large group of citizens, not just those who are

18 .

It’s saying that people who are

37

and

62

may vote; and people who are

12

may not. So it’s a statement about a large collection of numbers. In the second half of this section, we examine the mathematical notation for large collections of numbers like this.

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Figure 1.3.1. Alternative Video Lesson

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Subsection 1.3.1 Comparison Symbols

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In everyday language you can say something like “

8

is larger than

3

”. In mathematical writing, we have a shorthand notation for this: “

>

”. It’s used as follows:

8 > 3

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That short expression is read aloud as “

8

is greater than

3

”. The symbol “

>

” is called the greater-than symbol.

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Checkpoint 1.3.2.

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Use mathematical notation to write “ $11.5$ is greater than $4.2$ ”.
Use mathematical notation to write “age is greater than $20$ ”.

Explanation.

This is “ $11.5 > 4.2$ ”.
We can use the word age to represent age, and write $age > 20 .$ Or we could use an abbreviation like $a$ for age, and write $a > 20 .$ Or we could use $x$ as a generic variable, and write $x > 20 .$

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Remark 1.3.3.

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At some point in history, it was settled that “

>

” was a good symbol for “is greater than”. The tall side of the symbol is with the larger of the two numbers, and the small pointed side is with the smaller of the two numbers. One way to remember how this symbol works is to imagine it as an open mouth, and tell yourself that the mouth is hungry and it wants to eat the larger number.

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We have to be careful when negative numbers are used in a comparison. Is

- 8

greater or less than

- 3 ?

In one sense

- 8

is larger, because if you owe someone

8

dollars, that’s “more” than owing them

3

dollars. But the “

>

” symbol does not work that way. This symbol tells you which number is farther to the right on a number line. With that understanding,

- 3

is greater than

- 8 .

a number line with marks at -8, -3, and 8; text indicates the numbers to the left are lesser numbers, and numbers to the right are greater numbers; text indicates that -3 > -8 and 8 > -3 — 🔗
Figure 1.3.4. How the $>$ symbol works.

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Checkpoint 1.3.5.

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Use the

>

symbol to arrange the following numbers in order from greatest to least.

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(a)

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- 7.6 6 - 6 9.5 8

Explanation.

We can order these numbers by placing these numbers on a number line.

And so we see the answer is

9.5 > 8 > 6 > - 6 > - 7.6 .

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(b)

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- 5.2 3.14159 \frac{10}{3} 4.6 8

Explanation.

We can order these numbers by placing these numbers on a number line. Knowing or computing their decimals helps with this:

π \approx 3.141 \dots

and

\frac{10}{3} \approx 3.333 \dots .

And so we see the answer is

8 > 4.6 > \frac{10}{3} > 3.14159 > - 5.2 .

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The greater-than symbol has a close relative: the greater-than-or-equal-to symbol “

\geq

”. It means just like it sounds; the left number is either greater than or equal to the right number. Consider these examples, five of which are true and one of which is false.

\begin{aligned} 8 & \geq 3 & 3 & \geq - 8 & 3 & \geq 3 \\ 8 & > 3 & 3 & > - 8 & 3 & \overset{no}{>} 3 \end{aligned}

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While it may seem unhelpful to write

3 \geq 3

when you could write

3 = 3,

the “

\geq

” symbol is useful when at least one of the numbers in a comparison is not specific, like in these examples:

\begin{aligned} (hourly pay rate) & \geq (minimum wage) & (age of a voter) & \geq 18 \end{aligned}

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Sometimes you want to emphasize that one number is less than another number. For this, we have symbols that are reversed from

>

and

\geq .

The symbol “

<

” is the less-than symbol and it’s used like this:

3 < 8

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and read aloud as “

3

is less than

8

”.

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Table 6 gives the complete list of all six comparison symbols. We’ve only discussed three of them so far in this section, but you already know the equals symbol. The other two are the “less than or equal to” symbol, “

\leq

”, and the “not equal to” symbol, “

\neq

”.

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Table 1.3.6. Comparison Symbols

Symbol	Means	True	True	False
$=$	equals	$13 = 13$	$\frac{5}{4} = 1.25$	$5 \overset{no}{=} 6$
$>$	is greater than	$13 > 11$	$π > 3$	$9 \overset{no}{>} 9$
$\geq$	is greater than or equal to	$13 \geq 11$	$3 \geq 3$	$10.2 \overset{no}{\geq} 11.2$
$<$	is less than	$- 3 < 8$	$\frac{1}{2} < \frac{2}{3}$	$2 \overset{no}{<} - 2$
$\leq$	is less than or equal to	$- 3 \leq 8$	$3 \leq 3$	$\frac{4}{5} \overset{no}{\leq} \frac{3}{5}$
$\neq$	is not equal to	$10 \neq 20$	$\frac{1}{2} \neq 1.2$	$\frac{3}{8} \overset{no}{\neq} 0.375$

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Subsection 1.3.2 Set-Builder and Interval Notation

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If you write

(age of a voter) \geq 18

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and have a particular voter in mind, what is that person’s age? Maybe they are

18,

but maybe they are older. It’s helpful to use a variable

a

to represent age (in years) and then to visualize the possibilities with a number line.

$a number line, where the number 18 is marked; the portion of the number line to the right of 18 has a thick line overlaying it with an arrow pointing to the right; at 18, there is a left bracket character; text indicates the right area represents possibilities for age$

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Figure 1.3.7.

(age of a voter) \geq 18

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The shaded portion of the number line in Figure 7 is a mathematical interval. That means a collection of certain numbers with a “starting point” and a “ending point”. The interval above doesn’t really ever end, but we can say

\infty

(infinity) is the “ending point” in this situation. So this interval starts at

18

and “ends” at

\infty .

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The number line in Figure 7 is a visual representation of a collection of certain numbers. We have notations we can use to write down such collections of numbers.

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Definition 1.3.8. Set-Builder Notation.

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Set-builder notation attempts to say directly what condition needs to be met by numbers in the interval. We write set-builder notation like:

{x ∣ condition on x}

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and read it aloud as “the set of all

x

such that …”.

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For example,

{x ∣ x \geq 18}

is read aloud as “the set of all

x

such that

x

is greater than or equal to

18

”. The breakdown is as follows.

${x ∣ x \geq 18}$	the set of
${x ∣ x \geq 18}$	all $x$
${x ∣ x \geq 18}$	such that
${x ∣ x \geq 18}$	$x$ is greater than or equal to $18$

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Example 1.3.9.

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The set of all positive numbers is

{x ∣ x > 0},

but the set of all non-negative numbers is

{x ∣ x \geq 0} .

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The set of all possible Celsius temperatures for liquid water is

{x ∣ x > 0 and x < 100} .

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Checkpoint 1.3.10.

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For each interval expressed in the number lines, write the interval using set-builder notation.

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(a)

Explanation.

Since all numbers less than or equal to

2

are shaded, the set-builder notation is

{x ∣ x \leq 2} .

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(b)

Explanation.

Since all numbers less than to

2

are shaded, the set-builder notation is

{x ∣ x < 2} .

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(c)

Explanation.

Since all numbers greater than or equal to

2

are shaded, the set-builder notation is

{x ∣ x \geq 2} .

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Set-builder notation is useful, but there is an alternative for intervals that is less cumbersome.

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Definition 1.3.11. Interval Notation.

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Interval notation describes an interval by saying where it “starts” and “ends”. It can look like any of these four options:

(start, end) (start, end] [start, end) [start, end]

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In Figure 7, the interval starts at

18 .

Then it extends forever and has no end, so we use the

\infty

symbol for where this interval “ends”. And we write

[18, \infty) .

There is a subtlety about using the bracket “

[

” on one side and the parenthesis “

)

” on the other side. The bracket tells us that

18

is part of the interval and the parenthesis tells us that

\infty

is not part of the interval.

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Imagine if we wanted to describe all the numbers greater than

18,

including numbers like

18.01

but not including

18

itself. Then we would write

(18, \infty) .

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So there are four types of infinite intervals. Take note of the different uses of round parentheses and square brackets.

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(a, \infty)

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An open-infinite interval means all numbers

a

or greater, not including

a .

$a number line with a mark at a; the portion of the number line from a to the right has a thick line overlaying it; there is a left parenthesis at a and an arrow on the right pointing right$

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[a, \infty)

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A closed-infinite interval means all numbers

a

or greater, including

a .

$a number line with a mark at a; the portion of the number line from a to the right has a thick line overlaying it; there is a left bracket at a and an arrow on the right pointing right$

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(- \infty, a)

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An infinite-open interval means all numbers

a

or less, not including

a .

$a number line with a mark at a; the portion of the number line from a to the left has a thick line overlaying it; there is a right parenthesis at a and an arrow on the left pointing left$

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(- \infty, a]

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A infinite-closed interval means all numbers

a

or less, including

a .

$a number line with a mark at a; the portion of the number line from a to the left has a thick line overlaying it; there is a right bracket at a and an arrow on the left pointing left$

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Figure 1.3.12. Types of infinite intervals in interval notation

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Checkpoint 1.3.13. Interval Notation from Number Lines.

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For each interval expressed in the number lines, write the interval notation.

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(a)

Explanation.

The shaded interval “starts” at

- \infty

and ends at

2

(including

2

) so the interval notation is

(- \infty, 2] .

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(b)

Explanation.

The shaded interval “starts” at

- \infty

and ends at

2

(excluding

2

) so the interval notation is

(- \infty, 2) .

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(c)

Explanation.

The shaded interval starts at

2

(including

2

) and “ends” at

\infty,

so the interval notation is

[2, \infty) .

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Remark 1.3.14. Alternative Convention for Sketching Intervals.

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When graphing an interval, there is an alternative convention than you might see in other resources explaining algebra. This other convention uses open circles and filled-in circles. An open circle is used in place of a parenthesis, and a filled-in circle is used in place of a bracket, as in this example for the interval

[a, b) .

$a number line with marks at a and b; the portion of the number line between a and b has a thick line overlaying it; there is a hollow dot at a and solid dot at b$

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Reading Questions 1.3.3 Reading Questions

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1.

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How many inequality symbols are there?

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2.

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What is the difference between the interval

[3, \infty)

and the interval

(3, \infty) ?

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3.

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The expression

{x ∣ x \leq 10}

is an example of notation.

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Exercises 1.3.4 Exercises

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Prerequisite/Review Skills

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These exercises are only intended for students who are rusty with converting fractions to decimals. If you feel comfortable, proceed to Skills Practice.

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Fractions to Decimals.

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Without help from a calculator, convert the fraction to a decimal. If the decimal terminates, give its exact value. Otherwise round to at least three significant digits.

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