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Questions: Algebra BusinessCalculus
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Solve \(\displaystyle \frac{x}{x - 1} - \frac{1}{x - 4}=- \frac{3}{x^{2} - 5 x + 4}\).
Factoring the denominator on the right hand side gives \(\displaystyle \left(x - 4\right) \left(x - 1\right)\). This gives the LCD as \(\displaystyle \left(x - 4\right) \left(x - 1\right)\). Multiplying by the LCD gives \(\displaystyle x \left(x - 4\right) - x + 1 = -3\). Getting zero on one side gives \(\displaystyle x^{2} - 5 x + 4=0\). Factoring gives \(\displaystyle \left(x - 4\right) \left(x - 1\right)=0\). The two possible solutions are \(\displaystyle x = 1\) and \(\displaystyle x = 4\). Checking the possible solutions gives:
Since \(\displaystyle 1\) is zero of the denominator it is not in the domain and must be rejected as a solution. Since \(\displaystyle 4\) is zero of the denominator it is not in the domain and must be rejected as a solution. Therefore the equation has no solutions.
\begin{question}Solve $\frac{x}{x - 1} - \frac{1}{x - 4}=- \frac{3}{x^{2} - 5 x + 4}$.
\soln{9cm}{Factoring the denominator on the right hand side gives $\left(x - 4\right) \left(x - 1\right)$. This gives the LCD as $\left(x - 4\right) \left(x - 1\right)$. Multiplying by the LCD gives $x \left(x - 4\right) - x + 1 = -3$. Getting zero on one side gives $x^{2} - 5 x + 4=0$. Factoring gives $\left(x - 4\right) \left(x - 1\right)=0$. The two possible solutions are $x = 1$ and $x = 4$. Checking the possible solutions gives:\newline
Since $1$ is zero of the denominator it is not in the domain and must be rejected as a solution. Since $4$ is zero of the denominator it is not in the domain and must be rejected as a solution. Therefore the equation has no solutions. }
\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 \frac{x}{x - 1} - \frac{1}{x - 4}=- \frac{3}{x^{2} - 5 x + 4} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7Bx%7D%7Bx%20-%201%7D%20-%20%5Cfrac%7B1%7D%7Bx%20-%204%7D%3D-%20%5Cfrac%7B3%7D%7Bx%5E%7B2%7D%20-%205%20x%20%2B%204%7D%20" alt="LaTeX: \displaystyle \frac{x}{x - 1} - \frac{1}{x - 4}=- \frac{3}{x^{2} - 5 x + 4} " data-equation-content=" \displaystyle \frac{x}{x - 1} - \frac{1}{x - 4}=- \frac{3}{x^{2} - 5 x + 4} " /> . </p> </p><p> <p>Factoring the denominator on the right hand side gives <img class="equation_image" title=" \displaystyle \left(x - 4\right) \left(x - 1\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28x%20-%204%5Cright%29%20%5Cleft%28x%20-%201%5Cright%29%20" alt="LaTeX: \displaystyle \left(x - 4\right) \left(x - 1\right) " data-equation-content=" \displaystyle \left(x - 4\right) \left(x - 1\right) " /> . This gives the LCD as <img class="equation_image" title=" \displaystyle \left(x - 4\right) \left(x - 1\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28x%20-%204%5Cright%29%20%5Cleft%28x%20-%201%5Cright%29%20" alt="LaTeX: \displaystyle \left(x - 4\right) \left(x - 1\right) " data-equation-content=" \displaystyle \left(x - 4\right) \left(x - 1\right) " /> . Multiplying by the LCD gives <img class="equation_image" title=" \displaystyle x \left(x - 4\right) - x + 1 = -3 " src="/equation_images/%20%5Cdisplaystyle%20x%20%5Cleft%28x%20-%204%5Cright%29%20-%20x%20%2B%201%20%3D%20-3%20" alt="LaTeX: \displaystyle x \left(x - 4\right) - x + 1 = -3 " data-equation-content=" \displaystyle x \left(x - 4\right) - x + 1 = -3 " /> . Getting zero on one side gives <img class="equation_image" title=" \displaystyle x^{2} - 5 x + 4=0 " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%20-%205%20x%20%2B%204%3D0%20" alt="LaTeX: \displaystyle x^{2} - 5 x + 4=0 " data-equation-content=" \displaystyle x^{2} - 5 x + 4=0 " /> . Factoring gives <img class="equation_image" title=" \displaystyle \left(x - 4\right) \left(x - 1\right)=0 " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28x%20-%204%5Cright%29%20%5Cleft%28x%20-%201%5Cright%29%3D0%20" alt="LaTeX: \displaystyle \left(x - 4\right) \left(x - 1\right)=0 " data-equation-content=" \displaystyle \left(x - 4\right) \left(x - 1\right)=0 " /> . The two possible solutions are <img class="equation_image" title=" \displaystyle x = 1 " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%201%20" alt="LaTeX: \displaystyle x = 1 " data-equation-content=" \displaystyle x = 1 " /> and <img class="equation_image" title=" \displaystyle x = 4 " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%204%20" alt="LaTeX: \displaystyle x = 4 " data-equation-content=" \displaystyle x = 4 " /> . Checking the possible solutions gives:<br>
Since <img class="equation_image" title=" \displaystyle 1 " src="/equation_images/%20%5Cdisplaystyle%201%20" alt="LaTeX: \displaystyle 1 " data-equation-content=" \displaystyle 1 " /> is zero of the denominator it is not in the domain and must be rejected as a solution. Since <img class="equation_image" title=" \displaystyle 4 " src="/equation_images/%20%5Cdisplaystyle%204%20" alt="LaTeX: \displaystyle 4 " data-equation-content=" \displaystyle 4 " /> is zero of the denominator it is not in the domain and must be rejected as a solution. Therefore the equation has no solutions. </p> </p>