Problem 24 - Elementary Algebra

Question

Simplify each complex rational expression. $$ \frac{\frac{m}{m+1}-1}{\frac{m+1}{2}} $$

Step-by-Step Solution

Answer

Question: Simplify the complex rational expression: \(\frac{\frac{m}{m+1} - 1}{\frac{m+1}{2}}\). Answer: \(\frac{-2}{(m+1)^2}\).

Step 1: Simplify the numerator

Find a common denominator and subtract the fractions in the numerator. To do this, multiply 1 by the common denominator \((m+1)\) and subtract from the given fraction. $$ \frac{\frac{m}{m+1} - \frac{m+1}{m+1}}{\frac{m+1}{2}} $$

Step 2: Combine the terms in the numerator

Now that the fractions in the numerator have the same denominator, we can combine them by subtracting their numerators. $$ \frac{\frac{m-(m+1)}{m+1}}{\frac{m+1}{2}} $$

Step 3: Simplify the combined fraction in the numerator

Simplify the expression in the numerator. $$ \frac{\frac{-1}{m+1}}{\frac{m+1}{2}} $$

Step 4: Divide the numerator by the denominator

To divide fractions, we can multiply the numerator by the reciprocal of the denominator. $$ \frac{-1}{m+1} \cdot \frac{2}{m+1} $$

Step 5: Simplify the expression

Now, we can multiply the numerators and denominators, and simplify the result. $$ \frac{(-1) \cdot 2}{(m+1) \cdot (m+1)} = \frac{-2}{(m+1)^2} $$ The simplified complex rational expression is \(\frac{-2}{(m+1)^2}\).

Key Concepts

Simplifying Fractions

When working with fractions, simplifying them is all about reducing them to their simplest form. This means making the fraction as easy to work with as possible, without changing its value.
To simplify a fraction, you divide both the numerator (the top number) and the denominator (the bottom number) by their greatest common divisor (GCD).
In the context of complex rational expressions, this process involves dealing with fractions within fractions. It's crucial to simplify the inner fractions first to make the larger expression more manageable.

• Look for common factors in the numerator and denominator.
• Divide both numbers by their GCD.
• Repeat until no further simplification is possible.

This is an essential skill as it helps in making calculations straightforward and reduces errors in solving expressions.

Common Denominator

Finding a common denominator is a key step when adding or subtracting fractions, especially in complex rational expressions. A denominator is the bottom part of the fraction, and a common one means finding a shared denominator for two or more fractions to make computation easier.
When two fractions have a common denominator, you can simply add or subtract their numerators while keeping the denominator the same. This makes it simpler to perform operations within the fraction.
To find a common denominator:

• Identify the denominators of each fraction.
• Find the least common multiple (LCM) of these denominators.
• Adjust each fraction so that their denominators match, multiplying the top and bottom by necessary factors.

The process ensures that fractions are on the same terms, ready for further operations and simplifications.

Reciprocal

Reciprocal is a term you might come across often when playing with fractions. Simply put, the reciprocal of a fraction is what you get when you swap the numerator and the denominator.
The reciprocal is particularly useful when dividing fractions. Instead of actually dividing, which can be complex with fractions, you multiply by the reciprocal of the divisor. This operation flips the fraction, simplifying the division process significantly.

• A fraction like \( \frac{a}{b}\) has a reciprocal of \( \frac{b}{a}\).
• To divide, multiply by the reciprocal: \( \frac{x}{y} \div \frac{a}{b} = \frac{x}{y} \times \frac{b}{a}\).
• This method makes division easier to handle and more intuitive.

Using the reciprocal effectively decodes some of the complexities surrounding operations with fractions, making them much easier to solve.

Rational Expressions

Rational expressions are like fractions, but with polynomials in both the numerator and denominator. These expressions can seem daunting, but understanding them is key in tackling algebra problems related to fractions.
Working with rational expressions involves simplifying them, finding common denominators, and sometimes working with reciprocals. These expressions appear frequently in algebra and calculus, and knowing how to manipulate them is crucial.

• Simplify by factoring polynomials and cancelling common terms.
• Find a common denominator to combine terms or fractions.
• Use reciprocals to divide expressions.

Understanding rational expressions makes it easier to solve equations and inequalities, and helps in calculus where they recur in various forms.