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In mathematics, a reciprocal of a number is simply the number you need to multiply by to get the value of one. For any fraction \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \). This means you flip the numerator and the denominator. Understanding the concept of reciprocal is crucial when you are working with fractions and their operations.

• The reciprocal of a whole number, say 2, is \( \frac{1}{2} \).
• For a fraction like \( \frac{3}{4} \), the reciprocal would be \( \frac{4}{3} \).

To find the reciprocal, remember to swap the top and bottom of the fraction. When dealing with algebraic expressions, apply the same principle. For example, if you have an expression like \( \frac{x-7}{2(x-9)} \), its reciprocal would be \( \frac{2(x-9)}{x-7} \). This switching is essential for dividing fractions as it transforms your division problem into a multiplication one.

Dividing fractions might seem tricky at first, but it's straightforward once you know the steps. The key is to convert the division into multiplication using the reciprocal of the second fraction.

• The basic rule is: \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \).
• This means, instead of dividing by a fraction, you multiply by its reciprocal.

In the given exercise, you're dividing \( \frac{x+7}{2(x-9)} \) by \( \frac{x-7}{2(x-9)} \). Following the division rule by multiplying with the reciprocal, this becomes a straightforward expression to handle:
\[ \frac{x+7}{2(x-9)} \times \frac{2(x-9)}{x-7} \]
This method simplifies your work and helps avoid mistakes that are common when directly working with division of fractions.

Once you've successfully transformed a division problem into a multiplication problem, the next step is simplifying the expression. Simplifying involves canceling out terms that appear in both the numerator and the denominator.

• Look for common factors in the numerator and denominator.
• Cancel them out to simplify the expression and make it easier to solve.

In the exercise, notice that \(2(x-9)\) appears in both the numerator and the denominator:
\[ \frac{x+7}{2(x-9)} \times \frac{2(x-9)}{x-7} \]
By canceling \(2(x-9)\) from both, the expression simplifies to \( \frac{x+7}{x-7} \). Simplification reduces complexity and often makes the result more interpretable.