$$\frac{x+1}{2}=\frac{5}{x-2}$$

What is the solution set to the equation shown above?

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We can cross multiply first to remove the denominators.

$$\frac{x+1}{2}=\frac{5}{x-2}$$ $$(x+1)(x-2)=5(2)$$ $$x^2-x-2=10$$ $$x^2-x-12=0$$

Factoring, quadratic equation, and completing the square are all viable methods of obtaining the solution. Below is the factored form:

$$(x-4)(x+3)=0$$ $$x=4 \text{ and} -3$$

Neither of these solutions are excluded in the domain of the initial equation and are therefore valid solutions.

We can test the solutions in the answer choices. For example, we can exclude a few options since $x=-1$ does not satisfy the equation.

$$\frac{x+1}{2}=\frac{5}{x-2}$$ $$\frac{-1+1}{2}=\frac{5}{-1-2}$$ $$0 \neq -\frac{5}{3}$$

Testing other solutions would lead us to the correct answer choice.