Skip to main content
Mathematics LibreTexts

1.3: Inequalities and intervals

  • Page ID
    48949
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

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

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

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

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

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

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

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

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

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

    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


    This page titled 1.3: Inequalities and intervals is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.