Problem 23 - Algebra 1: Concepts and Skills

Question

SUBTRACTING RATIONAL EXPRESSIONS. Simplify the expression. $$ \frac{4 m}{m-2}-\frac{2 m+4}{m-2} $$

Step-by-Step Solution

Answer

The simplified form of the expression is \(\frac{2m - 4}{m - 2}\)

Step 1: Identify the Common Denominator

Observe the common denominator shared between both expressions, that is \(m-2\). It’s important to note that the denominators of the given expressions are the same. This already simplifies the process of subtraction for us.

Step 2: Combine Numerators

Since the denominators are identical, we can straightway subtract the numerators. Thus, we combine the numerators to create a single fraction: \[\frac{4m - (2m + 4)}{m - 2}\]

Step 3: Distribute Negative Sign

Next, distribute the negative sign to each term in the second numerator to obtain \[\frac{4m - 2m - 4}{m - 2}\]

Step 4: Simplify the Numerator

Now, simplify the numerator part by combining like terms which gives us \[\frac{2m -4}{m - 2}\]. This is our final simplified expression.

Key Concepts

Simplifying Rational Expressions

When simplifying rational expressions, the key is to reduce the expression to its simplest form. Rational expressions are fractions, but they involve polynomials in the numerator, denominator, or both. Simplifying them is akin to simplifying regular fractions. We aim to reduce common factors to most basic terms.

• Identify shared factors in the numerator and denominator.
• Factor out anything common, eliminating with caution if it cancels the entire term.

Always check to ensure the denominator is not zero, as division by zero is undefined. In the given exercise, after subtraction, we simplified by combining like terms, ending up with a numerator that was easy to manage. This sort of simplification helps in evaluating expressions more straightforwardly when plugging in actual values.

Common Denominator

A common denominator is a critical element when subtracting rational expressions. It refers to the same polynomial provided in the denominators of both fractions. Particularly, this makes the subtraction or addition significantly more manageable.
Receiving a problem where the denominators are the same, as in \( \frac{4m}{m-2} - \frac{2m+4}{m-2} \), reduces complexity. Here's why:

• The absence of multiple denominators negates the need for additional steps to find a lowest common denominator.
• You can directly proceed to add or subtract the numerators, making this a swift process.

For differing denominators, you would typically need to multiply each fraction by a factor that transforms the denominators to a common form.

Combining Like Terms

In math, combining like terms means summing up terms in an expression that share the same variable and exponent. It's an essential step when simplifying expressions because it makes them more concise and easier to interpret or solve.
With rational expressions, once you've combined the numerators as in \( \frac{4m - 2m - 4}{m - 2} \), look for terms that can be grouped together.

• In our example, \( 4m - 2m \) are like terms because they both include the variable \( m \).
• Subtract or add these coefficients (i.e., numbers in front of \( m \)) to consolidate without the variable changing.

This step simplifies our expression to \( \frac{2m - 4}{m - 2} \), making it streamlined and much clearer.

Negative Sign Distribution

Handling negative signs correctly is crucial in operations involving subtraction or equations in algebra. When subtracting rational expressions like \( \frac{4m}{m-2} - \frac{2m+4}{m-2} \), properly distributing the negative sign is critical. This can prevent errors in simplification.
Follow these steps:

• Write down each term in the expression, with parentheses around the second term when subtracting.
• Distribute the negative sign across all terms in the second expression’s numerator.

This practice ensures every term is properly accounted for, especially when subtracting multiple terms (e.g., changing \( 2m+4 \) to \( -2m-4 \) when the negative sign is distributed). It’s akin to changing the sign of each term. This simple step can reduce computation errors, making your final expression accurate and reliable.