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Solve \(\displaystyle x^{2} + 118\geq 6 - 23 x\)
Getting zero on the right hand side and factoring gives \(\displaystyle x^{2} + 23 x + 112=0\iff \left(x + 7\right) \left(x + 16\right)=0\). This gives the critical numbers as \(\displaystyle x=-7\) and \(\displaystyle x=-16\).
\begin{tikzpicture}[font=\Large]
\draw[latex-latex, very thick] (-18.2,0)--(-4.8,0) node[right]{$x$};
\foreach \x in { -18,-17,...,-5 }{
\draw (\x, .2) -- (\x,-.2) node[below]{$\x$};
}
\draw[dashed, thick, color=red] (-7, 0) -- (-7, 3.5);
\draw[dashed, thick, color=blue] (-16, 0) -- (-16, 3.5);
\draw (-20,1) node[left]{$x + 7$};
\draw (-20,2) node[left]{$x + 16$};
\draw (-20,3) node[left]{$f(x)$};
\draw (-18,1) node[left]{\color{red}$-7$};
\draw (-18,2) node[left]{\color{blue}$-16$};
\draw (-18,3) node[left]{zeros};
\draw (-17,3) node[left]{$+$};
\draw (-17,2) node[left]{$-$};
\draw (-17,1) node[left]{$-$};
\draw (-11.5,3) node[left]{$-$};
\draw (-11.5,2) node[left]{$+$};
\draw (-11.5,1) node[left]{$-$};
\draw (-6,3) node[left]{$+$};
\draw (-6,2) node[left]{$+$};
\draw (-6,1) node[left]{$+$};
\end{tikzpicture}
\begin{question}Solve $x^{2} + 118\geq 6 - 23 x$
\soln{9cm}{Getting zero on the right hand side and factoring gives $x^{2} + 23 x + 112=0\iff \left(x + 7\right) \left(x + 16\right)=0$. This gives the critical numbers as $x=-7$ and $x=-16$.
\begin{center}\begin{tikzpicture}[font=\Large]
\draw[latex-latex, very thick] (-18.2,0)--(-4.8,0) node[right]{$x$};
\foreach \x in { -18,-17,...,-5 }{
\draw (\x, .2) -- (\x,-.2) node[below]{$\x$};
}
\draw[dashed, thick, color=red] (-7, 0) -- (-7, 3.5);
\draw[dashed, thick, color=blue] (-16, 0) -- (-16, 3.5);
\draw (-20,1) node[left]{$x + 7$};
\draw (-20,2) node[left]{$x + 16$};
\draw (-20,3) node[left]{$f(x)$};
\draw (-18,1) node[left]{\color{red}$-7$};
\draw (-18,2) node[left]{\color{blue}$-16$};
\draw (-18,3) node[left]{zeros};
\draw (-17,3) node[left]{$+$};
\draw (-17,2) node[left]{$-$};
\draw (-17,1) node[left]{$-$};
\draw (-11.5,3) node[left]{$-$};
\draw (-11.5,2) node[left]{$+$};
\draw (-11.5,1) node[left]{$-$};
\draw (-6,3) node[left]{$+$};
\draw (-6,2) node[left]{$+$};
\draw (-6,1) node[left]{$+$};
\end{tikzpicture}
\end{center}
Checking the sign in each region gives the solution $(-\infty, -16 ] \cup [ -7 ,\infty)$.}
\end{question}
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\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Solve <img class="equation_image" title=" \displaystyle x^{2} + 118\geq 6 - 23 x " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%20%2B%20118%5Cgeq%206%20-%2023%20x%20" alt="LaTeX: \displaystyle x^{2} + 118\geq 6 - 23 x " data-equation-content=" \displaystyle x^{2} + 118\geq 6 - 23 x " /> </p> </p><p> <p>Getting zero on the right hand side and factoring gives <img class="equation_image" title=" \displaystyle x^{2} + 23 x + 112=0\iff \left(x + 7\right) \left(x + 16\right)=0 " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%20%2B%2023%20x%20%2B%20112%3D0%5Ciff%20%5Cleft%28x%20%2B%207%5Cright%29%20%5Cleft%28x%20%2B%2016%5Cright%29%3D0%20" alt="LaTeX: \displaystyle x^{2} + 23 x + 112=0\iff \left(x + 7\right) \left(x + 16\right)=0 " data-equation-content=" \displaystyle x^{2} + 23 x + 112=0\iff \left(x + 7\right) \left(x + 16\right)=0 " /> . This gives the critical numbers as <img class="equation_image" title=" \displaystyle x=-7 " src="/equation_images/%20%5Cdisplaystyle%20x%3D-7%20" alt="LaTeX: \displaystyle x=-7 " data-equation-content=" \displaystyle x=-7 " /> and <img class="equation_image" title=" \displaystyle x=-16 " src="/equation_images/%20%5Cdisplaystyle%20x%3D-16%20" alt="LaTeX: \displaystyle x=-16 " data-equation-content=" \displaystyle x=-16 " /> .
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Checking the sign in each region gives the solution <img class="equation_image" title=" \displaystyle (-\infty, -16 ] \cup [ -7 ,\infty) " src="/equation_images/%20%5Cdisplaystyle%20%28-%5Cinfty%2C%20-16%20%5D%20%5Ccup%20%5B%20-7%20%2C%5Cinfty%29%20" alt="LaTeX: \displaystyle (-\infty, -16 ] \cup [ -7 ,\infty) " data-equation-content=" \displaystyle (-\infty, -16 ] \cup [ -7 ,\infty) " /> .</p> </p>