1.3: Inequalities and intervals

Last updated
Save as PDF

Page ID: 48949

Thomas Tradler and Holly Carley
CUNY New York City College of Technology via New York City College of Technology at CUNY Academic Works

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$

There is an order relation on the set of real numbers:

$4< 9$ reads as $4$ is less than $9$,

$-3\leq 2$ reads as $-3$ is less than or equal to $2$,

$\dfrac{7}{6}> 1$ reads as $\dfrac{7}{6}$ is greater than $1$,

$2\geq -3$ reads as $2$ is greater than or equal to $-3$.

Example $\PageIndex{1}$

We have $2<3$, but $-2>-3$, which can be seen on the number line above.

Example $\PageIndex{2}$

We have $5\leq 5$ and $5\geq 5$. However the same is not true when using the symbol $<$. We write this as $5\nless 5$.

The set of all real numbers $x$ greater than or equal to some number $a$ and/or less than or equal to some number $b$ is denoted in different ways by the following chart:

Inequality notation	Number line	Interval notation
$a\leq x\leq b$	$clipboard_e2d5da3bc9abac4dc72586b1b4f4c88c3.png$	$[a,b]$
$a< x< b$	$clipboard_ef776bcd0d27aaed4806be926f2422ac4.png$	$(a,b)$
$a\leq x< b$	$clipboard_e2c4d21d9916a6a4391b0f6868ddd3b7e.png$	$[a,b)$
$a< x\leq b$	$clipboard_ee3b9bf7eecca4e44f461e15f70cca243.png$	$(a,b]$
$a\leq x$	$clipboard_e80bee8b845cb2f63905060f6cffa46fb.png$	$[a,\infty)$
$a< x$	$clipboard_eb054a3d76ed7af34b171bbc59b31be08.png$	$(a,\infty)$
$x\leq b$	$clipboard_eca76fe108f02ee849453ec2f3c448457.png$	$(-\infty,b]$
$x< b$	$clipboard_e5b981ec9777b5abd2eb12c4c85c6f91b.png$	$(-\infty,b)$

Formally, we define the interval $[a,b]$ to be the set of all real numbers $x$ such that $a\leq x \leq b$:

\[[a, b]=\{x \mid a \leq x \leq b\} \nonumber \]

There are similar definitions for the other intervals shown in the above table.

Note

Be sure to write the smaller number $a<b$ first when writing an interval $[a,b]$. For example, the interval $[5,3]=\left\{\begin{array}{l|l}
x \mid & 5 \leq x \leq 3\}
\end{array}\right.$ would be the empty set!

Example $\PageIndex{3}$

Graph the the inequality $\pi<x\leq 5$ on the number line and write it in interval notation.

Solution

On the number line: $clipboard_e2a14b770f9c970cba8cc243bb4019514.png$

Interval notation: $(\pi,5]$

Example $\PageIndex{4}$

Write the following interval as an inequality and in interval notation: $clipboard_e6ee687f45a795f577c96eb031c6315cd.png$

Solution

Inequality notation: $-3\leq x$

Interval notation: $[-3,\infty)$

Example $\PageIndex{5}$

Write the following interval as an inequality and in interval notation: $clipboard_edc567b9df61ba6120d465eea28e0ea59.png$

Solution

Inequality notation: & $x<2$

Interval notation: & $(-\infty,2)$

Note

In some texts round and square brackets are also used on the number line to depict an interval. For example the following displays the interval $[2,5)$.

$clipboard_e0074d9f841bf967f04e196f2f6cba271.png$

Inequality notation	Number line	Interval notation
\(a\leq x\leq b\)	$clipboard_e2d5da3bc9abac4dc72586b1b4f4c88c3.png$	\([a,b]\)
\(a< x< b\)	$clipboard_ef776bcd0d27aaed4806be926f2422ac4.png$	\((a,b)\)
\(a\leq x< b\)	$clipboard_e2c4d21d9916a6a4391b0f6868ddd3b7e.png$	\([a,b)\)
\(a< x\leq b\)	$clipboard_ee3b9bf7eecca4e44f461e15f70cca243.png$	\((a,b]\)
\(a\leq x\)	$clipboard_e80bee8b845cb2f63905060f6cffa46fb.png$	\([a,\infty)\)
\(a< x\)	$clipboard_eb054a3d76ed7af34b171bbc59b31be08.png$	\((a,\infty)\)
\(x\leq b\)	$clipboard_eca76fe108f02ee849453ec2f3c448457.png$	\((-\infty,b]\)
\(x< b\)	$clipboard_e5b981ec9777b5abd2eb12c4c85c6f91b.png$	\((-\infty,b)\)

Example \(\PageIndex{1}\)

Example \(\PageIndex{2}\)

Note

Example \(\PageIndex{3}\)

Example \(\PageIndex{4}\)

Example \(\PageIndex{5}\)

Note