Move the negative sign to the left.
\[\frac{4}{x+1}-\frac{1}{x}=-\frac{3}{x-4}\]
Multiply both sides by the Least Common Denominator: \(x(x+1)(x-4)\).
\[4x(x-4)-(x+1)(x-4)=-3x(x+1)\]
Simplify.
\[3{x}^{2}-13x+4=-3{x}^{2}-3x\]
Move all terms to one side.
\[3{x}^{2}-13x+4+3{x}^{2}+3x=0\]
Simplify \(3{x}^{2}-13x+4+3{x}^{2}+3x\) to \(6{x}^{2}-10x+4\).
\[6{x}^{2}-10x+4=0\]
Factor out the common term \(2\).
\[2(3{x}^{2}-5x+2)=0\]
Split the second term in \(3{x}^{2}-5x+2\) into two terms.
Multiply the coefficient of the first term by the constant term.
\[3\times 2=6\]
Ask: Which two numbers add up to \(-5\) and multiply to \(6\)?
Split \(-5x\) as the sum of \(-2x\) and \(-3x\).
\[3{x}^{2}-2x-3x+2\]
\[2(3{x}^{2}-2x-3x+2)=0\]
Factor out common terms in the first two terms, then in the last two terms.
\[2(x(3x-2)-(3x-2))=0\]
Factor out the common term \(3x-2\).
\[2(3x-2)(x-1)=0\]
Solve for \(x\).
Ask: When will \((3x-2)(x-1)\) equal zero?
When \(3x-2=0\) or \(x-1=0\)
Solve each of the 2 equations above.
\[x=\frac{2}{3},1\]
\[x=\frac{2}{3},1\]
Decimal Form: 0.666667, 1
x=2/3,1
