Problem 7 - Prealgebra and Introductory Algebra

Question

Simplify. $$\frac{3}{x+4}-\frac{10}{x+4}$$

Step-by-Step Solution

Answer

\(\frac{-7}{x+4}\)

Step 1: Identify Like Terms

Here you observe that both fractions have the same denominator, \(x+4\). When the denominators are the same, the fractions can be combined directly by subtracting the numerators.

Step 2: Subtract Numerators

Subtract the fractions by subtracting the numerators. That is, \(\frac{3}{x+4}-\frac{10}{x+4} = \frac{3-10}{x+4}\).

Step 3: Simplify the Numerator

Finally, simplify the numerator to get \(\frac{-7}{x+4}\).

Key Concepts

Understanding Like Terms

When simplifying algebraic fractions, identifying 'like terms' is a crucial first step. Like terms are terms that have the same variables raised to the same power. In the given problem, we look at the denominators of \(\frac{3}{x+4}\) and \(\frac{10}{x+4}\). They both contain the same term, \(x+4\), which means they are like terms. Because they're the same, the algebraic fractions can be combined into a single fraction. This is akin to adding or subtracting apples; you can only directly combine them if they're the same type of fruit. In algebra, only like terms can be combined in this direct way.

Recognizing like terms allows us to combine or separate expressions with the assurance that we're manipulating the equation appropriately, keeping the balance of the equation intact. To illustrate, consider that you have 3 apples and you subtract 10 apples; this scenario is represented by our fractions having the same denominator. Just like apples in this example, we can directly subtract the quantities because we're dealing with the same units, or in our math problem, the same algebraic terms.

Subtracting Numerators

Subtracting numerators is a straightforward process when you're working with algebraic fractions that have like terms as their denominators. Imagine you have a pie divided into equal slices and two people have taken some slices. To find out how many slices are left, you would subtract the number of slices one person took from the slices the other person has. Similarly, in the given problem \(\frac{3}{x+4} - \frac{10}{x+4}\), we simply subtract the numerators, the 'slices' in this case, while keeping the denominator, the 'whole pie', unchanged.

The operation follows the basic arithmetic property: \(\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\). Every time you subtract numerators, you're essentially combining those parts to see what remains. In the given problem, we subtract 10 from 3, considering they are parts of the same whole, denoted by the common denominator \(x+4\).

Simplifying the Numerator

After subtracting the numerators, you often have to simplify the numerator to get the fraction in its simplest form. This means reducing the expression to its most basic terms, where no further subtraction can be made. In the problem \(\frac{3-10}{x+4}\), the numerator simplifies to \(\frac{-7}{x+4}\) when we subtract 10 from 3.

Why is it important to simplify? Simplifying an expression makes it easier to understand and handle in later calculations. It's like cleaning up after cooking; you'll have a clearer workspace for whatever comes next. The process of simplifying can also reveal properties about the expression that aren't obvious in its complex form. For example, a simplified numerator can make it apparent whether a fraction can be further reduced, can help us identify zeros of the function, and can aid in graphing the algebraic fraction if needed.

In practice, simplifying the numerator might involve combining terms, reducing the terms by common factors, or even factoring polynomials, all aimed at achieving the most elementary form of the expression.