Answer:
x = 8
Step-by-step explanation:
There are several ways you can solve a proportion like this. Most of the algebraic solutions involve what amounts to multiplying both sides by (4x/3). There are also graphical solutions and solutions that rely on relations that make fractions be equal.
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cross multiplication
The method of solution usually recommended is to "clear fractions" and then solve the resulting one-step linear equation. To "clear fractions," multiply both sides of the equation by the least common denominator: 4x.
[tex](4x)\dfrac{3}{4}=(4x)\dfrac{6}{x}\\\\3\cdot x=4\cdot6\qquad\text{looks like "cross multiplication"}[/tex]
We describe this as "cross-multiplication" because it looks like each numerator has been multiplied by the opposite denominator.
This "one-step" linear equation is solved by dividing by the coefficient of x, which is 3:
[tex]\dfrac{3x}{3}=\dfrac{4\cdot6}{3}\\\\\boxed{x=8}\qquad\text{simplify}[/tex]
Note that we could have done these operations in one step. The two steps, multiply by 4x and divide by 3, are identical to the one step, multiply by (4x/3).
[tex]\left(\dfrac{4x}{3}\right)\cdot\dfrac{3}{4}=\left(\dfrac{4x}{3}\right)\cdot\dfrac{6}{x}\\\\\boxed{x=8}\qquad\text{simplify}[/tex]
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compare fractions
We can see the solution almost immediately if we rewrite the fractions so they have the same numerator.
[tex]\dfrac{3}{4}=\dfrac{6}{x}\\\\\dfrac{3\cdot2}{4\cdot2}=\dfrac{6}{x}\\\\\dfrac{6}{8}=\dfrac{6}{x}\ \Longrightarrow\ \boxed{x=8}[/tex]
The same thing happens if we multiply the equation by the inverse of either side.
[tex]\dfrac{3}{4}=\dfrac{6}{x}\ \Longrightarrow\ \left(\dfrac{4}{3}\right)\dfrac{3}{4}=\left(\dfrac{4}{3}\right)\dfrac{6}{x}\ \Longrightarrow\ 1=\dfrac{8}{x}\ \Longrightarrow\ \boxed{x=8}\\\\\textsf{ or ...}\\\\\dfrac{3}{4}=\dfrac{6}{x}\ \Longrightarrow\ \left(\dfrac{x}{6}\right)\dfrac{3}{4}=\left(\dfrac{x}{6}\right)\dfrac{6}{x}\ \Longrightarrow\ \dfrac{x}{8}=1\ \Longrightarrow\ \boxed{x=8}[/tex]
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graphical solution
If the equation is rewritten to a form that has a value of 0 at the solution, then the x-intercept of the graph will be the solution to the equation.
3/4 = 6/x . . . . . . . . original equation
6/x -3/4 = 0 ⇒ f(x) = 0
f(x) = 6/x -3/4
The graph shows the solution to f(x) = 0 is x=8.
