Problem 96 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the operation and simplify. $$\frac{x+7}{2(x-9)} \div \frac{x-7}{2(x-9)}$$

Step-by-Step Solution

Answer

The simplified form of the operation is \( (x+7)/(x-7) \).

Step 1: Recognize the Division

Here the task is to divide one fraction by another. When dividing fractions, it's important to remember that dividing by a fraction is the same as multiplying by its reciprocal.

Step 2: Write Division as Multiplication

Rewrite the division as multiplication by taking the reciprocal of the second fraction. \[x+7/2(x-9)\] divided by \[x-7/2(x-9)\] could also be written as \[x+7/2(x-9)\] multiplied by \[2(x-9)/(x-7)\]. The expression would then become: \[= \frac{x+7}{2(x-9)} \times \frac{2(x-9)}{x-7}\].

Step 3: Simplify the Expression

Now you can cancel out the common factors in the numerator and the denominator. Here, \[2(x-9)\] is common in both the numerator and the denominator, so it can be cancelled out. You will be left with \[(x+7)/(x-7)\]. This is the simplified form of the division.

Key Concepts

Reciprocal

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.

Fraction Division

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.

Simplifying Expressions

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.