Algebraic fractions

Simplify \[ \begin{align} &\frac{x+2}{x^2-3x+2} - \frac{1}{x^2-2x} \end{align} \ \]

Put everything over a common denominator

\[ \begin{align} \frac{x+2}{x^2-3x+2} - \frac{1}{x^2-2x} &= \frac{x+2}{(x-1)(x-2)} - \frac{1}{x(x-2)} \\ &= \frac{x\times (x+2) - (x-1) \times 1}{x(x-1)(x-2)} \end{align} \ \]

Simplify numerator and denominator

\[ \begin{align} &= \frac{x^2+x+1}{x(x-1)(x-2)} \end{align} \ \]

You will gain most benefit if you work through each problem first and then come back and check your work against the solution.

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Now You Try:

Simplify \[ \begin{align} &\frac{x}{x-2} - \frac{x-1}{x^2-5x+6} \end{align} \ \]