Answer :
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.
After solving [tex]\frac{x}{2}=\frac{3x+4}{2x}[/tex] we get value of x as 4 and -1. The correct option is A) x=4, -1
Given expression is,
[tex]\frac{x}{2}=\frac{3x+4}{2x}[/tex]
Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.
x(2x)=2(3x+4)
Solve the equation for x
Simplify x(2x).
[tex]2x^2[/tex] =2(3x+4)
Simplify: 2(3x+4)
[tex]2x^2[/tex]=6x+8
Subtract 6x from both sides of the equation.
[tex]2x^2[/tex]−6x=8
Subtract 8 from both sides of the equation.
[tex]2x^2[/tex]−6x−8=0
Factor the left side of the equation.
2(x−4)(x+1)=0
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.
x−4=0
x+1=0
Set x−4 equal to 0 and solve for x.
x=4
Set x+1 equal to 0 and solve for x.
x=−1
The final solution is all the values that make 2(x−4)(x+1)=0.
x=4,−1
