Problem 54 - Elementary Algebra

Question

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{x}{x-2}}{\frac{x+4}{x+3}} $$

Step-by-Step Solution

Answer

\( \frac{x(x+3)}{(x-2)(x+4)} \)

Step 1 - Rewrite the fraction

Rewrite the given complex fraction as a division of two fractions: \ \( \frac{x}{x-2} \ \frac{x+4}{x+3} \)

Step 2 - Multiply by the reciprocal

Convert the division into a multiplication by taking the reciprocal of the divisor: \ \( \frac{x}{x-2} \times \frac{x+3}{x+4} \)

Step 3 - Simplify the expression

Now, multiply the numerators together and the denominators together: \ \( \frac{x \times (x+3)}{(x-2) \times (x+4)} \)

Step 4 - Final simplified form

The simplified expression is: \ \( \frac{x(x+3)}{(x-2)(x+4)} \)

Key Concepts

multiplying fractions

Multiplying fractions is a straightforward process that makes use of the numerators (the top parts) and denominators (the bottom parts) of the given fractions.
To multiply fractions, simply multiply the numerators together and the denominators together.
For example, given two fractions, \(\frac{a}{b} \) and \(\frac{c}{d} \), their product is \(\frac{a \times c}{b \times d} \).
This concept is crucial when simplifying complex fractions.
In our exercise, when converting the division problem into multiplication by the reciprocal, we multiplied: \(\frac{x}{x-2} \times \frac{x+3}{x+4} \).
Here’s how to do it step-by-step:

• Multiply the numerators: \(x \times (x + 3) \)
• Multiply the denominators: \((x - 2) \times (x + 4) \)

This results in the simplified fraction: \(\frac{x(x+3)}{(x-2)(x+4)} \).

reciprocal of a fraction

The reciprocal of a fraction is simply flipping the fraction.
This means that the numerator becomes the denominator and the denominator becomes the numerator.
For example, the reciprocal of \(\frac{a}{b} \) is \(\frac{b}{a} \).
This is important because when dividing by a fraction, you actually multiply by its reciprocal.
In our exercise, we were given \(\frac{\frac{x}{x-2}}{\frac{x+4}{x+3}} \).
To simplify this, we took the reciprocal of \(\frac{x+4}{x+3} \), which is \(\frac{x+3}{x+4} \), and then multiplied:

• Reciprocal of \(\frac{x+4}{x+3} \) is \(\frac{x+3}{x+4} \)
• Multiply \(\frac{x}{x-2} \) by the reciprocal: \(\frac{x}{x-2} \times \frac{x+3}{x+4} \)

This method helps transform a division problem into an easier multiplication problem.

algebraic expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators.
In our exercise, we worked with expressions like \(\frac{x}{x-2} \) and \(\frac{x+4}{x+3} \).
Understanding and simplifying these expressions is vital for solving complex fractions.
Here are a few things to remember:

• Variables represent unknown values
• Expressions can be fractions, sums, differences, products, or quotients of numbers and variables
• You can simplify algebraic expressions by combining like terms or using algebraic rules

In our complex fraction \(\frac{\frac{x}{x-2}}{\frac{x+4}{x+3}} \), we simplified the numerator and denominator by treating them as separate algebraic fractions before multiplying.
Finally, combining these simplified forms results in \(\frac{x(x+3)}{(x-2)(x+4)} \), which is much easier to work with.