Calculate: (x)/(x-9)-(8)/(x+2)=(0x+99)/(x^2-7x-18)

Expression: $\frac{ x }{ x-9 }-\frac{ 8 }{ x+2 }=\frac{ 0x+99 }{ {x}^{2}-7x-18 }$

Determine the defined range

$\begin{array} { l }\frac{ x }{ x-9 }-\frac{ 8 }{ x+2 }=\frac{ 0x+99 }{ {x}^{2}-7x-18 },& \begin{array} { l }x≠9,& x≠-2\end{array}\end{array}$

Any expression multiplied by $0$ equals $0$

$\frac{ x }{ x-9 }-\frac{ 8 }{ x+2 }=\frac{ 0+99 }{ {x}^{2}-7x-18 }$

Removing $0$ doesn't change the value, so remove it from the expression

$\frac{ x }{ x-9 }-\frac{ 8 }{ x+2 }=\frac{ 99 }{ {x}^{2}-7x-18 }$

Move the expression to the left-hand side and change its sign

$\frac{ x }{ x-9 }-\frac{ 8 }{ x+2 }-\frac{ 99 }{ {x}^{2}-7x-18 }=0$

Write $-7x$ as a difference

$\frac{ x }{ x-9 }-\frac{ 8 }{ x+2 }-\frac{ 99 }{ {x}^{2}+2x-9x-18 }=0$

Factor out $x$ from the expression

$\frac{ x }{ x-9 }-\frac{ 8 }{ x+2 }-\frac{ 99 }{ x \times \left( x+2 \right)-9x-18 }=0$

Factor out $-9$ from the expression

$\frac{ x }{ x-9 }-\frac{ 8 }{ x+2 }-\frac{ 99 }{ x \times \left( x+2 \right)-9\left( x+2 \right) }=0$

Factor out $x+2$ from the expression

$\frac{ x }{ x-9 }-\frac{ 8 }{ x+2 }-\frac{ 99 }{ \left( x+2 \right) \times \left( x-9 \right) }=0$

Write all numerators above the least common denominator $\left( x+2 \right) \times \left( x-9 \right)$

$\frac{ x \times \left( x+2 \right)-8\left( x-9 \right)-99 }{ \left( x+2 \right) \times \left( x-9 \right) }=0$

Distribute $x$ through the parentheses

$\frac{ {x}^{2}+2x-8\left( x-9 \right)-99 }{ \left( x+2 \right) \times \left( x-9 \right) }=0$

Distribute $-8$ through the parentheses

$\frac{ {x}^{2}+2x-8x+72-99 }{ \left( x+2 \right) \times \left( x-9 \right) }=0$

Collect like terms

$\frac{ {x}^{2}-6x+72-99 }{ \left( x+2 \right) \times \left( x-9 \right) }=0$

Calculate the difference

$\frac{ {x}^{2}-6x-27 }{ \left( x+2 \right) \times \left( x-9 \right) }=0$

Write $-6x$ as a difference

$\frac{ {x}^{2}+3x-9x-27 }{ \left( x+2 \right) \times \left( x-9 \right) }=0$

Factor out $x$ from the expression

$\frac{ x \times \left( x+3 \right)-9x-27 }{ \left( x+2 \right) \times \left( x-9 \right) }=0$

Factor out $-9$ from the expression

$\frac{ x \times \left( x+3 \right)-9\left( x+3 \right) }{ \left( x+2 \right) \times \left( x-9 \right) }=0$

Factor out $x+3$ from the expression

$\frac{ \left( x+3 \right) \times \left( x-9 \right) }{ \left( x+2 \right) \times \left( x-9 \right) }=0$

Cancel out the common factor $x-9$

$\frac{ x+3 }{ x+2 }=0$

When the quotient of expressions equals $0$, the numerator has to be $0$

$x+3=0$

Move the constant to the right-hand side and change its sign

$\begin{array} { l }x=-3,& \begin{array} { l }x≠9,& x≠-2\end{array}\end{array}$

Check if the solution is in the defined range

$x=-3$