Question

1. \\(\frac{x}{x - 1} + \frac{x + 7}{x^2 - 1} - \frac{x - 2}{x + 1}\\)

Explanation:

Step1: Factor the denominator

Notice that \(x^2 - 1=(x - 1)(x + 1)\) by the difference of squares formula \(a^2 - b^2=(a - b)(a + b)\) where \(a = x\) and \(b = 1\). So the expression becomes \(\frac{x}{x - 1}+\frac{x + 7}{(x - 1)(x + 1)}-\frac{x - 2}{x + 1}\).

Step2: Find a common denominator

The common denominator for the three fractions is \((x - 1)(x + 1)\). Rewrite each fraction with this common denominator:

• For \(\frac{x}{x - 1}\), multiply numerator and denominator by \((x + 1)\): \(\frac{x(x + 1)}{(x - 1)(x + 1)}\)
• For \(\frac{x + 7}{(x - 1)(x + 1)}\), it already has the common denominator: \(\frac{x + 7}{(x - 1)(x + 1)}\)
• For \(\frac{x - 2}{x + 1}\), multiply numerator and denominator by \((x - 1)\): \(\frac{(x - 2)(x - 1)}{(x - 1)(x + 1)}\)

Step3: Combine the fractions

Now that all fractions have the same denominator, we can combine them:

$$
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$$

Step4: Simplify the numerator

Factor out a 5 from the numerator: \(\frac{5(x + 1)}{(x - 1)(x + 1)}\)

Step5: Cancel common factors

Cancel out the common factor of \((x + 1)\) in the numerator and denominator (note that \(x
eq - 1\) to avoid division by zero): \(\frac{5}{x - 1}\)

Answer:

\(\frac{5}{x - 1}\)