Guzman Bob
05/08/2024 · Elementary School
11 Solve \( \frac{2}{5-x}+\frac{3}{x+7}=1 \)
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Tutor-Verified Answer
Step-by-step Solution
Solve the equation \( \frac{2}{5-x}+\frac{3}{x+7}=1 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(\frac{2}{5-x}+\frac{3}{x+7}=1\)
- step1: Find the domain:
\(\frac{2}{5-x}+\frac{3}{x+7}=1,x \in \left(-\infty,-7\right)\cup \left(-7,5\right)\cup \left(5,+\infty\right)\)
- step2: Multiply both sides of the equation by LCD:
\(\left(\frac{2}{5-x}+\frac{3}{x+7}\right)\left(5-x\right)\left(x+7\right)=1\times \left(5-x\right)\left(x+7\right)\)
- step3: Simplify the equation:
\(-x+29=-2x+35-x^{2}\)
- step4: Move the expression to the left side:
\(-x+29-\left(-2x+35-x^{2}\right)=0\)
- step5: Calculate:
\(x-6+x^{2}=0\)
- step6: Factor the expression:
\(\left(-2+x\right)\left(3+x\right)=0\)
- step7: Separate into possible cases:
\(\begin{align}&-2+x=0\\&3+x=0\end{align}\)
- step8: Solve the equation:
\(\begin{align}&x=2\\&x=-3\end{align}\)
- step9: Check if the solution is in the defined range:
\(\begin{align}&x=2\\&x=-3\end{align},x \in \left(-\infty,-7\right)\cup \left(-7,5\right)\cup \left(5,+\infty\right)\)
- step10: Find the intersection:
\(\begin{align}&x=2\\&x=-3\end{align}\)
- step11: Rewrite:
\(x_{1}=-3,x_{2}=2\)
The solutions to the equation \( \frac{2}{5-x}+\frac{3}{x+7}=1 \) are \( x=-3 \) and \( x=2 \).
Quick Answer
The solutions are \( x=-3 \) and \( x=2 \).
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