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{\dsum}{\displaystyle\sum\limits} $

$ \newcommand{\dint}{\displaystyle\int\limits} $

$ \newcommand{\dlim}{\displaystyle\lim\limits} $

$ \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}}$

$ \newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$ \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$ \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vectorC}[1]{\textbf{#1}} $

$ \newcommand{\vectorD}[1]{\overrightarrow{#1}} $

$ \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} $

$ \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} $

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$\newcommand{\longvect}{\overrightarrow}$

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$\newcommand{\ket}[1]{\left| #1 \right>}$
$\newcommand{\bra}[1]{\left< #1 \right|}$
$\newcommand{\braket}[2]{\left< #1 \vphantom{#2} \right| \left. #2 \vphantom{#1} \right>}$
$\newcommand{\braopket}[3]{\left< #1 \vphantom{#2}\vphantom{#3} \right| #2 \vphantom{#1}\vphantom{#3} \left| #3 \vphantom{#1}\vphantom{#2} \right>}$
$\newcommand{\qmvec}[1]{\mathbf{\vec{#1}}}$
$\newcommand{\op}[1]{\hat{\mathbf{#1}}}$
$\newcommand{\expect}[1]{\langle #1 \rangle}$
$\newcommand{\dfn}[1]{\emph{\textbf{#1}}}$

$\newcommand{\avec}{\mathbf a}$ $\newcommand{\bvec}{\mathbf b}$ $\newcommand{\cvec}{\mathbf c}$ $\newcommand{\dvec}{\mathbf d}$ $\newcommand{\dtil}{\widetilde{\mathbf d}}$ $\newcommand{\evec}{\mathbf e}$ $\newcommand{\fvec}{\mathbf f}$ $\newcommand{\nvec}{\mathbf n}$ $\newcommand{\pvec}{\mathbf p}$ $\newcommand{\qvec}{\mathbf q}$ $\newcommand{\svec}{\mathbf s}$ $\newcommand{\tvec}{\mathbf t}$ $\newcommand{\uvec}{\mathbf u}$ $\newcommand{\vvec}{\mathbf v}$ $\newcommand{\wvec}{\mathbf w}$ $\newcommand{\xvec}{\mathbf x}$ $\newcommand{\yvec}{\mathbf y}$ $\newcommand{\zvec}{\mathbf z}$ $\newcommand{\rvec}{\mathbf r}$ $\newcommand{\mvec}{\mathbf m}$ $\newcommand{\zerovec}{\mathbf 0}$ $\newcommand{\onevec}{\mathbf 1}$ $\newcommand{\real}{\mathbb R}$ $\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$ $\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$ $\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$ $\newcommand{\laspan}[1]{\text{Span}\{#1\}}$ $\newcommand{\bcal}{\cal B}$ $\newcommand{\ccal}{\cal C}$ $\newcommand{\scal}{\cal S}$ $\newcommand{\wcal}{\cal W}$ $\newcommand{\ecal}{\cal E}$ $\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$ $\newcommand{\gray}[1]{\color{gray}{#1}}$ $\newcommand{\lgray}[1]{\color{lightgray}{#1}}$ $\newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\row}{\text{Row}}$ $\newcommand{\col}{\text{Col}}$ $\renewcommand{\row}{\text{Row}}$ $\newcommand{\nul}{\text{Nul}}$ $\newcommand{\var}{\text{Var}}$ $\newcommand{\corr}{\text{corr}}$ $\newcommand{\len}[1]{\left|#1\right|}$ $\newcommand{\bbar}{\overline{\bvec}}$ $\newcommand{\bhat}{\widehat{\bvec}}$ $\newcommand{\bperp}{\bvec^\perp}$ $\newcommand{\xhat}{\widehat{\xvec}}$ $\newcommand{\vhat}{\widehat{\vvec}}$ $\newcommand{\uhat}{\widehat{\uvec}}$ $\newcommand{\what}{\widehat{\wvec}}$ $\newcommand{\Sighat}{\widehat{\Sigma}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\newcommand{\amp}{&}$ $\definecolor{fillinmathshade}{gray}{0.9}$

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

Search

Text Color

Text Size

Margin Size

Font Type

Example \(\PageIndex{1}\)

Example \(\PageIndex{2}\)

Note

Example \(\PageIndex{3}\)

Example \(\PageIndex{4}\)

Example \(\PageIndex{5}\)

Note