Saltar al contenido principal
LibreTexts Español

1.2: El valor absoluto

  1. Última actualización
  2. Guardar como PDF
  • Page ID
    117751

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

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

    \( \newcommand{\dsum}{\displaystyle\sum\limits} \)

    \( \newcommand{\dint}{\displaystyle\int\limits} \)

    \( \newcommand{\dlim}{\displaystyle\lim\limits} \)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    \(\newcommand{\longvect}{\overrightarrow}\)

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

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

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

    El valor absoluto de un número real\(c\), denotado por\(|c|\) el número no negativo que es igual en magnitud (o tamaño) a\(c\), es decir, es el número resultante de ignorar el signo:

    \ [|c|=\ left\ {\ begin {array} {cl}
    c, &\ text {if} c\ text {es positivo o cero}\\
    -c, &\ text {if} c\ text {es negativo}
    \ end {array}\ right. \ nonumber\]

    Ejemplo\(\PageIndex{1}\)

    \(|-4|=4\)

    Ejemplo\(\PageIndex{2}\)

    \(|12|=12\)

    Ejemplo\(\PageIndex{3}\)

    \(|-3.523|=3.523\)

    Ejemplo\(\PageIndex{4}\)

    ¿Para\(x\) qué números reales tienes\(|x|=3\)?

    Solución

    Desde\(|3|=3\) y\(|-3|=3\), vemos que hay dos soluciones,\(x=3\) o\(x=-3\).

    El conjunto de soluciones es\(S=\{-3,3\}\).

    Ejemplo\(\PageIndex{5}\)

    |x|=5] Resolver para\(x\):\(|x|=5\)

    Solución

    \(x=5\)o\(x=-5\). El conjunto de soluciones es\(S=\{-5,5\}\).

    Ejemplo\(\PageIndex{6}\)

    |x|=-7] Resolver para\(x\):\(|x|=-7\).

    Solución

    Tenga en cuenta\(|7|=7\) eso\(|-7|=7\) y para que estos no puedan dar ninguna solución. En efecto, no hay soluciones, ya que el valor absoluto es siempre no negativo. El conjunto de soluciones es el conjunto vacío\(S=\{\}\).

    Ejemplo\(\PageIndex{7}\)

    Resolver para\(x\):\(|x|=0\).

    Solución

    Ya que\(-0=0\), sólo hay una solución,\(x=0\). Por lo tanto,\(S=\{0\}\).

    Ejemplo\(\PageIndex{8}\)

    Resolver para\(x\):\(|x+2|=6\).

    Solución

    Ya que el valor absoluto de\(x+2\) es\(6\), vemos que\(x+2\) tiene que ser cualquiera\(6\) o\(-6\).

    Evaluamos cada caso,

    \ [\ begin {array} {l|l}
    \ text {ya sea} x+2=6, &\ text {o} x+2=-6\\
    \ Longrightarrow x=6-2, &\ LongRightarrow x=-6-2\
    \ LongRightarrow x=4, &\ LongRightarrow x=-8
    \ end {array}\ nonumber\]

    El conjunto de soluciones es\(S=\{-8,4\}\).

    Ejemplo\(\PageIndex{9}\)

    Resolver para\(x\):\(|3x-4|=5\)

    Solución

    \ [\ begin {array} {l|l}
    \ text {O} 3 x-4=5 &\ text {o} 3 x-4=-5\\
    \ Longrightarrow 3 x=9 &\ Longrightarrow 3 x=-1\
    \ LongRightarrow x=3 &\ Rightarrow x=-\ frac {1} {3}
    \ end {array}\ nonumber\]

    El conjunto de soluciones es\(S=\{-\frac 1 3,3\}\).

    Ejemplo\(\PageIndex{10}\)

    Resolver para\(x\):\(-2\cdot |12+3x|=-18\)

    Solución

    Dividiendo ambos lados por\(-2\) da\(|12+3x|=9\). Con esto, tenemos los dos casos

    \ [\ begin {array} {l|l}
    \ text {O} 12+3 x=9 &\ text {o} 12+3 x=-9\\
    \ Longrightarrow 3 x=-3 &\ Longrightarrow 3 x=-21\
    \ LongRightarrow x=-1 &\ LongRightarrow x=-7
    \ end {array}\ nonumber\]

    El conjunto de soluciones es\(S=\{-7,-1\}\).

    This page titled 1.2: El valor absoluto was last modified on Sun, 30 Oct 2022 16:25:12 GMT and 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.