Problem 1 - Elementary and Intermediate Algebra

Question

Fill in the blanks. \(\frac{x}{x-7}\) and \(\frac{1}{x-7}\) have like denominators. \(\frac{x+5}{x-7}\) and \(\frac{4 x}{x+7}\) have ____ denominators.

Step-by-Step Solution

Answer

Unlike

Step 1: Identifying Like Denominators

First, recognize that two fractions have like denominators if the expressions in the denominators are identical. Here we have two pairs of fractions: \(\frac{x}{x-7}\) and \(\frac{1}{x-7}\) with like denominators because both denominators are \(x-7\).

Step 2: Examining New Fractions

Look at the new pair of fractions: \(\frac{x+5}{x-7}\) and \(\frac{4x}{x+7}\). The denominators here are \(x-7\) and \(x+7\).

Step 3: Comparing New Denominators

Comparing the denominators \(x-7\) and \(x+7\), we see they are different because one is \(x-7\) and the other is \(x+7\).

Step 4: Conclusion

Fractions \(\frac{x+5}{x-7}\) and \(\frac{4x}{x+7}\) do not have the same denominators. Thus, they have unlike denominators.

Key Concepts

Understanding Like Denominators

When we talk about "like denominators," we're referring to fractions that share the exact same denominator. To illustrate, consider the fractions \(\frac{x}{x-7}\) and \(\frac{1}{x-7}\). Both of these fractions have the denominator \(x-7\), which means they have like denominators. This commonality is crucial because it simplifies the process of adding or subtracting fractions.
When fractions have like denominators, you only need to perform the operation on the numerators. The denominator remains the same, allowing for straightforward calculations. If you were to add the fractions above, the result would be \(\frac{x+1}{x-7}\).
Recognizing and working with like denominators is a foundational skill when dealing with fractions in both everyday math and more advanced algebra.

Understanding Unlike Denominators

Fractions with unlike denominators are those where the denominator values are different. A good example from our exercise is \(\frac{x+5}{x-7}\) and \(\frac{4x}{x+7}\). Here, the denominators \(x-7\) and \(x+7\) are not the same, thus classified as unlike.
When dealing with fractions that have unlike denominators, you must first find a common denominator before performing operations such as addition or subtraction. This process typically involves finding the least common multiple of the denominators.
Let's say you want to add \(\frac{1}{4}\) and \(\frac{1}{6}\). You need to convert these to have a common denominator, like 12, to combine them easily as \(\frac{3}{12} + \frac{2}{12} = \frac{5}{12}\).
Understanding unlike denominators leads to mastering the art of finding common denominators, a vital step in simplifying complex fraction expressions.

Exploring Algebraic Fractions

Algebraic fractions are fractions where the numerator and/or the denominator contain algebraic expressions. These differ from simple fractions that typically involve only numbers. For example, \(\frac{x}{x-7}\), \(\frac{1}{x-7}\), and \(\frac{x+5}{x-7}\) are all algebraic fractions.
Manipulating algebraic fractions follows similar rules as numerical fractions, but you also need to consider the variables and algebraic expressions involved. When simplifying or finding common denominators, it's essential to treat these expressions carefully.
Here's a quick tip: When dealing with addition or subtraction of algebraic fractions with unlike denominators, you must first determine a common algebraic expression that can serve as a shared denominator. This might involve factoring or finding expressions that involve the least common multiple of the terms present in the denominators.

• Always simplify your algebraic fractions when possible, by reducing common factors.
• Watch out for undefined values where the denominator of a fraction could equal zero. For example, the expression \(x-7\) in the denominator would make the fraction undefined if \(x=7\).

By mastering algebraic fractions, you open doors to solving more complex algebraic equations and enhance your mathematical capabilities.