Problem 20 - Technical Mathematics with Calculus

Question

Combine and simplify. Try some by calculator. $$\frac{7}{x+2}-\frac{5}{x+2}$$

Step-by-Step Solution

Answer

\(\frac{2}{x+2}\)

Step 1: Identify Common Denominators

Look at the two fractions and notice that they have a common denominator \(x+2\).

Step 2: Combine the Numerators

Since the denominators are the same, the numerators can be combined by subtracting. Perform the subtraction \(7-5\) to get \(2\).

Step 3: Write the Final Simplified Result

The result of the combination is a single fraction with the common denominator: \[\frac{2}{x+2}\].

Key Concepts

Common Denominators in Algebraic Fractions

Understanding common denominators is crucial when working with algebraic fractions. A denominator is the bottom part of a fraction that indicates the number of equal parts the whole is divided into. When you have two or more fractions with the same denominator, you have what's called a 'common denominator'.

This is especially helpful when you need to add, subtract, or compare fractions. Having a common denominator ensures that the fractions are speaking the same 'mathematical language', allowing them to be easily combined or compared. Imagine trying to combine pieces of two puzzles; if they're from different puzzles (different denominators), they won't fit together.

With common denominators, you can directly compare or combine the numerators (the top numbers) because the pieces fit together perfectly. Always remember to check for common denominators before proceeding with operations on fractions to simplify your work and avoid unnecessary calculations.

Combining Numerators

The combining of numerators is a straightforward process once a common denominator is established. The numerator represents the number of parts you have, and when fractions have the same denominator, all you need to do is perform the operation instructed (like addition or subtraction) on the numerators directly.

For example, if you're dealing with the subtraction or addition of fractions with a common denominator, you simply subtract or add the numerators while keeping the denominator the same. It's similar to saying, 'If I have 7 apples and I give away 5 apples, how many am I left with?' The concept is the same, even when the numerators are algebraic expressions instead of simple numbers.

However, remember to always simplify your final answer when possible, by reducing the fraction to its lowest terms or expressing the result in terms of common factors. This neatness in calculation can make a significant difference in the clarity and correctness of your results.

Fraction Subtraction

When it comes to fraction subtraction, the process might seem complicated, but it can be made simple with a good grasp of the common denominator concept. Essentially, when subtracting fractions with the same denominator, you maintain the denominator and subtract the first numerator from the second.

In mathematical terms, if you have two fractions \(\frac{a}{c} - \frac{b}{c}\), you would subtract \( b \) from \( a \) to find the new numerator, making the result \(\frac{a - b}{c}\). Subtraction of fractions with different denominators involves a few more steps, such as finding a common denominator, but that's a discussion for another time.

It's also essential when subtracting fractions to pay attention to the signs and order of the numerators, as they can affect your answer. And as with any mathematical operation, always simplify your answer when possible to get the most reduced form of the fraction.