Answer:
A) x = 4, -1
Step-by-step explanation:
You want the solutions to the rational equation ...
[tex]\dfrac{x}{2}=\dfrac{3x+4}{2x}[/tex]
Trial and error
We can rearrange this equation to make it easy to try the different answer choices.
[tex]\dfrac{x}{2}=\dfrac{3x}{2x}+\dfrac{4}{2x}=\dfrac{3}{2}+\dfrac{2}{x}\\\\\\x=3+\dfrac{4}{x}\qquad\text{multiply by 2}[/tex]
Trying the first answer choice, we have ...
4 = 3 +4/4 . . . . true
-1 = 3 +4/(-1) . . . . true
The values of answer choice A satisfy the equation.
Quadratic
Multiplying the equation by 2x gives ...
x² = 3x +4
x² -3x -4 = 0 . . . . put in standard form
(x -4)(x +1) = 0 . . . . factor
x = 4, -1 . . . . . . . . . values that make the factors zero
The solutions are x = 4, -1, matching choice A.
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Additional comment
Subtracting the right-side expression gives an equation of the form f(x)=0. That is, the solutions will be the x-intercepts of the graph of f(x). The attachment shows these solutions. In general, a graphing calculator can find solutions to problems like this pretty easily.
