A rational function can have a variable like "x" in the numerator AND the denominator. When this happens, there are some special rules and properties to consider.

Solve the following for \(x\):

\[\frac{ x-14 } { x - 7 } = 1 + \frac{ 14 } { x - 28}. \]

Solve for \(x\):

\[ \frac{5}{x } + \frac{ 3x + 8 }{ x^2 - 8 x } = \frac{7 x + 8 }{ x^2- 8 x} . \]

Solve the following for \(x:\)

\[\frac{ 28 } { x^2 - 4x } = 1 + \frac{ 7 } { x - 4}. \]

How many solutions are there for

\[ \frac { 18 }{ x^2 + 18x } + \frac{ 18 }{ x^2 + 54x + 648 } = - \frac{ 1}{ 9 } ? \]

Solve the following for \(x:\)

\[\frac{ 6 } { x - 6 } = \frac{ x } { 8 } - 1. \]

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