# Evaluate: (x+1)/(x-1)=(-8)/(x+3)+(8)/(x^2+2x-3)

## Expression: $\frac{ x+1 }{ x-1 }=\frac{ -8 }{ x+3 }+\frac{ 8 }{ {x}^{2}+2x-3 }$

Determine the defined range

$\begin{array} { l }\frac{ x+1 }{ x-1 }=\frac{ -8 }{ x+3 }+\frac{ 8 }{ {x}^{2}+2x-3 },& \begin{array} { l }x≠1,& x≠-3\end{array}\end{array}$

Use $\frac{ -a }{ b }=\frac{ a }{ -b }=-\frac{ a }{ b }$ to rewrite the fraction

$\frac{ x+1 }{ x-1 }=-\frac{ 8 }{ x+3 }+\frac{ 8 }{ {x}^{2}+2x-3 }$

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

$\frac{ x+1 }{ x-1 }+\frac{ 8 }{ x+3 }-\frac{ 8 }{ {x}^{2}+2x-3 }=0$

Write $2x$ as a difference

$\frac{ x+1 }{ x-1 }+\frac{ 8 }{ x+3 }-\frac{ 8 }{ {x}^{2}+3x-x-3 }=0$

Factor out $x$ from the expression

$\frac{ x+1 }{ x-1 }+\frac{ 8 }{ x+3 }-\frac{ 8 }{ x \times \left( x+3 \right)-x-3 }=0$

Factor out the negative sign from the expression

$\frac{ x+1 }{ x-1 }+\frac{ 8 }{ x+3 }-\frac{ 8 }{ x \times \left( x+3 \right)-\left( x+3 \right) }=0$

Factor out $x+3$ from the expression

$\frac{ x+1 }{ x-1 }+\frac{ 8 }{ x+3 }-\frac{ 8 }{ \left( x+3 \right) \times \left( x-1 \right) }=0$

Write all numerators above the least common denominator $\left( x+3 \right) \times \left( x-1 \right)$

$\frac{ \left( x+3 \right) \times \left( x+1 \right)+8\left( x-1 \right)-8 }{ \left( x+3 \right) \times \left( x-1 \right) }=0$

Simplify the expression

$\frac{ {x}^{2}+x+3x+3+8\left( x-1 \right)-8 }{ \left( x+3 \right) \times \left( x-1 \right) }=0$

Distribute $8$ through the parentheses

$\frac{ {x}^{2}+x+3x+3+8x-8-8 }{ \left( x+3 \right) \times \left( x-1 \right) }=0$

Collect like terms

$\frac{ {x}^{2}+12x+3-8-8 }{ \left( x+3 \right) \times \left( x-1 \right) }=0$

Calculate the difference

$\frac{ {x}^{2}+12x-13 }{ \left( x+3 \right) \times \left( x-1 \right) }=0$

Write $12x$ as a difference

$\frac{ {x}^{2}+13x-x-13 }{ \left( x+3 \right) \times \left( x-1 \right) }=0$

Factor out $x$ from the expression

$\frac{ x \times \left( x+13 \right)-x-13 }{ \left( x+3 \right) \times \left( x-1 \right) }=0$

Factor out the negative sign from the expression

$\frac{ x \times \left( x+13 \right)-\left( x+13 \right) }{ \left( x+3 \right) \times \left( x-1 \right) }=0$

Factor out $x+13$ from the expression

$\frac{ \left( x+13 \right) \times \left( x-1 \right) }{ \left( x+3 \right) \times \left( x-1 \right) }=0$

Cancel out the common factor $x-1$

$\frac{ x+13 }{ x+3 }=0$

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

$x+13=0$

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

$\begin{array} { l }x=-13,& \begin{array} { l }x≠1,& x≠-3\end{array}\end{array}$

Check if the solution is in the defined range

$x=-13$

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