Solve 1/(x-2)*(x-3)-2/(x-1)*(x-3)+1/(x-1)*(x-2)

$$\frac{ 1 }{ (x-2) \times (x-3) } - \frac{ 2 }{ (x-1) \times (x-3) } + \frac{ 1 }{ (x-1) \times (x-2) }$$

Evaluate

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Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x-3\right) and \left(x-1\right)\left(x-3\right) is \left(x-3\right)\left(x-2\right)\left(x-1\right). Multiply \frac{1}{\left(x-2\right)\left(x-3\right)} times \frac{x-1}{x-1}. Multiply \frac{2}{\left(x-1\right)\left(x-3\right)} times \frac{x-2}{x-2}.

$$\frac{x-1}{\left(x-3\right)\left(x-2\right)\left(x-1\right)}-\frac{2\left(x-2\right)}{\left(x-3\right)\left(x-2\right)\left(x-1\right)}+\frac{1}{\left(x-1\right)\left(x-2\right)}$$

Since \frac{x-1}{\left(x-3\right)\left(x-2\right)\left(x-1\right)} and \frac{2\left(x-2\right)}{\left(x-3\right)\left(x-2\right)\left(x-1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x-1-2\left(x-2\right)}{\left(x-3\right)\left(x-2\right)\left(x-1\right)}+\frac{1}{\left(x-1\right)\left(x-2\right)}$$

Do the multiplications in x-1-2\left(x-2\right).

$$\frac{x-1-2x+4}{\left(x-3\right)\left(x-2\right)\left(x-1\right)}+\frac{1}{\left(x-1\right)\left(x-2\right)}$$

Combine like terms in x-1-2x+4.

$$\frac{-x+3}{\left(x-3\right)\left(x-2\right)\left(x-1\right)}+\frac{1}{\left(x-1\right)\left(x-2\right)}$$

Extract the negative sign in -x+3.

$$\frac{-\left(x-3\right)}{\left(x-3\right)\left(x-2\right)\left(x-1\right)}+\frac{1}{\left(x-1\right)\left(x-2\right)}$$

Cancel out x-3 in both numerator and denominator.

$$\frac{-1}{\left(x-2\right)\left(x-1\right)}+\frac{1}{\left(x-1\right)\left(x-2\right)}$$

Since \frac{-1}{\left(x-2\right)\left(x-1\right)} and \frac{1}{\left(x-1\right)\left(x-2\right)} have the same denominator, add them by adding their numerators. Add -1 and 1 to get 0.

$$\frac{0}{\left(x-2\right)\left(x-1\right)}$$

Zero divided by any non-zero term gives zero.

$$0$$

Factor

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