\(\text{www.the}\beta\text{etafunction.com}\)
Home
Login
Questions: Algebra BusinessCalculus
Please login to create an exam or a quiz.
Solve the inequality \(\displaystyle \frac{2}{x^{2} - 1}<\frac{3}{x^{2} + 9 x + 8}\)
Getting zero on one side and factoring gives \(\displaystyle - \frac{3}{\left(x + 1\right) \left(x + 8\right)} + \frac{2}{\left(x - 1\right) \left(x + 1\right)}< 0\). This gives the least common denominator as \(\displaystyle \left(x - 1\right) \left(x + 1\right) \left(x + 8\right)\). Building each fraction to get the common denominator gives \(\displaystyle \frac{2 x + 16 - (3 x - 3)}{\left(x - 1\right) \left(x + 1\right) \left(x + 8\right)} < 0\). Simplifying gives \(\displaystyle \frac{19 - x}{\left(x - 1\right) \left(x + 1\right) \left(x + 8\right)}<0\). The inequality can change signs at the zeros of the numerator, \(\displaystyle \left\{19\right\}\), or the zeros of the denominator \(\displaystyle \left\{-8, -1, 1\right\}\). Making a sign chart gives: This gives the solution \(\displaystyle \left(-\infty, -8\right) \cup \left(-1, 1\right) \cup \left(19, \infty\right)\)
\begin{question}Solve the inequality $\frac{2}{x^{2} - 1}<\frac{3}{x^{2} + 9 x + 8}$
\soln{9cm}{Getting zero on one side and factoring gives $- \frac{3}{\left(x + 1\right) \left(x + 8\right)} + \frac{2}{\left(x - 1\right) \left(x + 1\right)}< 0$. This gives the least common denominator as $\left(x - 1\right) \left(x + 1\right) \left(x + 8\right)$. Building each fraction to get the common denominator gives $\frac{2 x + 16 - (3 x - 3)}{\left(x - 1\right) \left(x + 1\right) \left(x + 8\right)} < 0$. Simplifying gives $\frac{19 - x}{\left(x - 1\right) \left(x + 1\right) \left(x + 8\right)}<0$. The inequality can change signs at the zeros of the numerator, $\left\{19\right\}$, or the zeros of the denominator $\left\{-8, -1, 1\right\}$. Making a sign chart gives: \begin{tikzpicture}
\draw[latex-latex, thick] (-3, 0) -- (7, 0);
\draw(-1, 0.2) -- (-1, -0.2);
\draw (-1,-0.2) node[below]{$-8$};
\draw (-2,.2) node[above]{$-$};
\draw(1, 0.2) -- (1, -0.2);
\draw (1,-0.2) node[below]{$-1$};
\draw (0,.2) node[above]{$+$};
\draw(3, 0.2) -- (3, -0.2);
\draw (3,-0.2) node[below]{$1$};
\draw (2,.2) node[above]{$-$};
\draw(5, 0.2) -- (5, -0.2);
\draw (5,-0.2) node[below]{$19$};
\draw (6,.2) node[above]{$-$};
\draw (4,.2) node[above]{$+$};
\end{tikzpicture}
This gives the solution $\left(-\infty, -8\right) \cup \left(-1, 1\right) \cup \left(19, \infty\right)$}
\end{question}
\documentclass{article}
\usepackage{tikz}
\usepackage{amsmath}
\usepackage[margin=2cm]{geometry}
\usepackage{tcolorbox}
\newcounter{ExamNumber}
\newcounter{questioncount}
\stepcounter{questioncount}
\newenvironment{question}{{\noindent\bfseries Question \arabic{questioncount}.}}{\stepcounter{questioncount}}
\renewcommand{\labelenumi}{{\bfseries (\alph{enumi})}}
\newif\ifShowSolution
\newcommand{\soln}[2]{%
\ifShowSolution%
\noindent\begin{tcolorbox}[colframe=blue,title=Solution]#2\end{tcolorbox}\else%
\vspace{#1}%
\fi%
}%
\newcommand{\hideifShowSolution}[1]{%
\ifShowSolution%
%
\else%
#1%
\fi%
}%
\everymath{\displaystyle}
\ShowSolutiontrue
\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Solve the inequality <img class="equation_image" title=" \displaystyle \frac{2}{x^{2} - 1}<\frac{3}{x^{2} + 9 x + 8} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B2%7D%7Bx%5E%7B2%7D%20-%201%7D%3C%5Cfrac%7B3%7D%7Bx%5E%7B2%7D%20%2B%209%20x%20%2B%208%7D%20" alt="LaTeX: \displaystyle \frac{2}{x^{2} - 1}<\frac{3}{x^{2} + 9 x + 8} " data-equation-content=" \displaystyle \frac{2}{x^{2} - 1}<\frac{3}{x^{2} + 9 x + 8} " /> </p> </p><p> <p>Getting zero on one side and factoring gives <img class="equation_image" title=" \displaystyle - \frac{3}{\left(x + 1\right) \left(x + 8\right)} + \frac{2}{\left(x - 1\right) \left(x + 1\right)}< 0 " src="/equation_images/%20%5Cdisplaystyle%20-%20%5Cfrac%7B3%7D%7B%5Cleft%28x%20%2B%201%5Cright%29%20%5Cleft%28x%20%2B%208%5Cright%29%7D%20%2B%20%5Cfrac%7B2%7D%7B%5Cleft%28x%20-%201%5Cright%29%20%5Cleft%28x%20%2B%201%5Cright%29%7D%3C%200%20" alt="LaTeX: \displaystyle - \frac{3}{\left(x + 1\right) \left(x + 8\right)} + \frac{2}{\left(x - 1\right) \left(x + 1\right)}< 0 " data-equation-content=" \displaystyle - \frac{3}{\left(x + 1\right) \left(x + 8\right)} + \frac{2}{\left(x - 1\right) \left(x + 1\right)}< 0 " /> . This gives the least common denominator as <img class="equation_image" title=" \displaystyle \left(x - 1\right) \left(x + 1\right) \left(x + 8\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28x%20-%201%5Cright%29%20%5Cleft%28x%20%2B%201%5Cright%29%20%5Cleft%28x%20%2B%208%5Cright%29%20" alt="LaTeX: \displaystyle \left(x - 1\right) \left(x + 1\right) \left(x + 8\right) " data-equation-content=" \displaystyle \left(x - 1\right) \left(x + 1\right) \left(x + 8\right) " /> . Building each fraction to get the common denominator gives <img class="equation_image" title=" \displaystyle \frac{2 x + 16 - (3 x - 3)}{\left(x - 1\right) \left(x + 1\right) \left(x + 8\right)} < 0 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B2%20x%20%2B%2016%20-%20%283%20x%20-%203%29%7D%7B%5Cleft%28x%20-%201%5Cright%29%20%5Cleft%28x%20%2B%201%5Cright%29%20%5Cleft%28x%20%2B%208%5Cright%29%7D%20%3C%200%20" alt="LaTeX: \displaystyle \frac{2 x + 16 - (3 x - 3)}{\left(x - 1\right) \left(x + 1\right) \left(x + 8\right)} < 0 " data-equation-content=" \displaystyle \frac{2 x + 16 - (3 x - 3)}{\left(x - 1\right) \left(x + 1\right) \left(x + 8\right)} < 0 " /> . Simplifying gives <img class="equation_image" title=" \displaystyle \frac{19 - x}{\left(x - 1\right) \left(x + 1\right) \left(x + 8\right)}<0 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B19%20-%20x%7D%7B%5Cleft%28x%20-%201%5Cright%29%20%5Cleft%28x%20%2B%201%5Cright%29%20%5Cleft%28x%20%2B%208%5Cright%29%7D%3C0%20" alt="LaTeX: \displaystyle \frac{19 - x}{\left(x - 1\right) \left(x + 1\right) \left(x + 8\right)}<0 " data-equation-content=" \displaystyle \frac{19 - x}{\left(x - 1\right) \left(x + 1\right) \left(x + 8\right)}<0 " /> . The inequality can change signs at the zeros of the numerator, <img class="equation_image" title=" \displaystyle \left\{19\right\} " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%5C%7B19%5Cright%5C%7D%20" alt="LaTeX: \displaystyle \left\{19\right\} " data-equation-content=" \displaystyle \left\{19\right\} " /> , or the zeros of the denominator <img class="equation_image" title=" \displaystyle \left\{-8, -1, 1\right\} " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%5C%7B-8%2C%20-1%2C%201%5Cright%5C%7D%20" alt="LaTeX: \displaystyle \left\{-8, -1, 1\right\} " data-equation-content=" \displaystyle \left\{-8, -1, 1\right\} " /> . Making a sign chart gives: <?xml version="1.0" encoding="UTF-8"?>
<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="284.261pt" height="38.911pt" viewBox="0 0 284.261 38.911" version="1.1">
<defs>
<g>
<symbol overflow="visible" id="glyph0-0">
<path style="stroke:none;" d=""/>
</symbol>
<symbol overflow="visible" id="glyph0-1">
<path style="stroke:none;" d="M 6.5625 -2.296875 C 6.734375 -2.296875 6.921875 -2.296875 6.921875 -2.5 C 6.921875 -2.6875 6.734375 -2.6875 6.5625 -2.6875 L 1.171875 -2.6875 C 1 -2.6875 0.828125 -2.6875 0.828125 -2.5 C 0.828125 -2.296875 1 -2.296875 1.171875 -2.296875 Z M 6.5625 -2.296875 "/>
</symbol>
<symbol overflow="visible" id="glyph1-0">
<path style="stroke:none;" d=""/>
</symbol>
<symbol overflow="visible" id="glyph1-1">
<path style="stroke:none;" d="M 1.625 -4.5625 C 1.171875 -4.859375 1.125 -5.1875 1.125 -5.359375 C 1.125 -5.96875 1.78125 -6.390625 2.484375 -6.390625 C 3.203125 -6.390625 3.84375 -5.875 3.84375 -5.15625 C 3.84375 -4.578125 3.453125 -4.109375 2.859375 -3.765625 Z M 3.078125 -3.609375 C 3.796875 -3.984375 4.28125 -4.5 4.28125 -5.15625 C 4.28125 -6.078125 3.40625 -6.640625 2.5 -6.640625 C 1.5 -6.640625 0.6875 -5.90625 0.6875 -4.96875 C 0.6875 -4.796875 0.703125 -4.34375 1.125 -3.875 C 1.234375 -3.765625 1.609375 -3.515625 1.859375 -3.34375 C 1.28125 -3.046875 0.421875 -2.5 0.421875 -1.5 C 0.421875 -0.453125 1.4375 0.21875 2.484375 0.21875 C 3.609375 0.21875 4.5625 -0.609375 4.5625 -1.671875 C 4.5625 -2.03125 4.453125 -2.484375 4.0625 -2.90625 C 3.875 -3.109375 3.71875 -3.203125 3.078125 -3.609375 Z M 2.078125 -3.1875 L 3.3125 -2.40625 C 3.59375 -2.21875 4.0625 -1.921875 4.0625 -1.3125 C 4.0625 -0.578125 3.3125 -0.0625 2.5 -0.0625 C 1.640625 -0.0625 0.921875 -0.671875 0.921875 -1.5 C 0.921875 -2.078125 1.234375 -2.71875 2.078125 -3.1875 Z M 2.078125 -3.1875 "/>
</symbol>
<symbol overflow="visible" id="glyph1-2">
<path style="stroke:none;" d="M 2.9375 -6.375 C 2.9375 -6.625 2.9375 -6.640625 2.703125 -6.640625 C 2.078125 -6 1.203125 -6 0.890625 -6 L 0.890625 -5.6875 C 1.09375 -5.6875 1.671875 -5.6875 2.1875 -5.953125 L 2.1875 -0.78125 C 2.1875 -0.421875 2.15625 -0.3125 1.265625 -0.3125 L 0.953125 -0.3125 L 0.953125 0 C 1.296875 -0.03125 2.15625 -0.03125 2.5625 -0.03125 C 2.953125 -0.03125 3.828125 -0.03125 4.171875 0 L 4.171875 -0.3125 L 3.859375 -0.3125 C 2.953125 -0.3125 2.9375 -0.421875 2.9375 -0.78125 Z M 2.9375 -6.375 "/>
</symbol>
<symbol overflow="visible" id="glyph1-3">
<path style="stroke:none;" d="M 4.078125 -2.296875 L 6.859375 -2.296875 C 7 -2.296875 7.1875 -2.296875 7.1875 -2.5 C 7.1875 -2.6875 7 -2.6875 6.859375 -2.6875 L 4.078125 -2.6875 L 4.078125 -5.484375 C 4.078125 -5.625 4.078125 -5.8125 3.875 -5.8125 C 3.671875 -5.8125 3.671875 -5.625 3.671875 -5.484375 L 3.671875 -2.6875 L 0.890625 -2.6875 C 0.75 -2.6875 0.5625 -2.6875 0.5625 -2.5 C 0.5625 -2.296875 0.75 -2.296875 0.890625 -2.296875 L 3.671875 -2.296875 L 3.671875 0.5 C 3.671875 0.640625 3.671875 0.828125 3.875 0.828125 C 4.078125 0.828125 4.078125 0.640625 4.078125 0.5 Z M 4.078125 -2.296875 "/>
</symbol>
<symbol overflow="visible" id="glyph1-4">
<path style="stroke:none;" d="M 3.65625 -3.171875 L 3.65625 -2.84375 C 3.65625 -0.515625 2.625 -0.0625 2.046875 -0.0625 C 1.875 -0.0625 1.328125 -0.078125 1.0625 -0.421875 C 1.5 -0.421875 1.578125 -0.703125 1.578125 -0.875 C 1.578125 -1.1875 1.34375 -1.328125 1.125 -1.328125 C 0.96875 -1.328125 0.671875 -1.25 0.671875 -0.859375 C 0.671875 -0.1875 1.203125 0.21875 2.046875 0.21875 C 3.34375 0.21875 4.5625 -1.140625 4.5625 -3.28125 C 4.5625 -5.96875 3.40625 -6.640625 2.515625 -6.640625 C 1.96875 -6.640625 1.484375 -6.453125 1.0625 -6.015625 C 0.640625 -5.5625 0.421875 -5.140625 0.421875 -4.390625 C 0.421875 -3.15625 1.296875 -2.171875 2.40625 -2.171875 C 3.015625 -2.171875 3.421875 -2.59375 3.65625 -3.171875 Z M 2.421875 -2.40625 C 2.265625 -2.40625 1.796875 -2.40625 1.5 -3.03125 C 1.3125 -3.40625 1.3125 -3.890625 1.3125 -4.390625 C 1.3125 -4.921875 1.3125 -5.390625 1.53125 -5.765625 C 1.796875 -6.265625 2.171875 -6.390625 2.515625 -6.390625 C 2.984375 -6.390625 3.3125 -6.046875 3.484375 -5.609375 C 3.59375 -5.28125 3.640625 -4.65625 3.640625 -4.203125 C 3.640625 -3.375 3.296875 -2.40625 2.421875 -2.40625 Z M 2.421875 -2.40625 "/>
</symbol>
</g>
</defs>
<g id="surface1">
<path style="fill:none;stroke-width:0.79701;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -80.379406 -0.00134375 L 193.765125 -0.00134375 " transform="matrix(1,0,0,-1,85.438,19.151)"/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 0.398438 19.152344 C 1.777344 19.410156 4.023438 20.1875 5.578125 21.09375 L 5.578125 17.207031 C 4.023438 18.113281 1.777344 18.890625 0.398438 19.152344 "/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 283.867188 19.152344 C 282.484375 18.890625 280.238281 18.113281 278.683594 17.207031 L 278.683594 21.09375 C 280.238281 20.1875 282.484375 19.410156 283.867188 19.152344 "/>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -28.348156 5.670531 L -28.348156 -5.669313 " transform="matrix(1,0,0,-1,85.438,19.151)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-1" x="50.726" y="34.76"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-1" x="58.475" y="34.76"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-1" x="24.871" y="9.132"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 28.347156 5.670531 L 28.347156 -5.669313 " transform="matrix(1,0,0,-1,85.438,19.151)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-1" x="107.419" y="34.76"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-2" x="115.168" y="34.76"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-3" x="81.563" y="9.132"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 85.038562 5.670531 L 85.038562 -5.669313 " transform="matrix(1,0,0,-1,85.438,19.151)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-2" x="167.986" y="34.76"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-1" x="138.256" y="9.132"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 141.733875 5.670531 L 141.733875 -5.669313 " transform="matrix(1,0,0,-1,85.438,19.151)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-2" x="222.189" y="34.76"/>
<use xlink:href="#glyph1-4" x="227.1703" y="34.76"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-1" x="251.642" y="9.132"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-3" x="194.949" y="9.132"/>
</g>
</g>
</svg>
This gives the solution <img class="equation_image" title=" \displaystyle \left(-\infty, -8\right) \cup \left(-1, 1\right) \cup \left(19, \infty\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28-%5Cinfty%2C%20-8%5Cright%29%20%5Ccup%20%5Cleft%28-1%2C%201%5Cright%29%20%5Ccup%20%5Cleft%2819%2C%20%5Cinfty%5Cright%29%20" alt="LaTeX: \displaystyle \left(-\infty, -8\right) \cup \left(-1, 1\right) \cup \left(19, \infty\right) " data-equation-content=" \displaystyle \left(-\infty, -8\right) \cup \left(-1, 1\right) \cup \left(19, \infty\right) " /> </p> </p>