Problem 15 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{9}{x-3}+\frac{9}{3-x}\)

Step-by-Step Solution

Answer

The simplified result is 0.

Step 1: Understand the Expressions

Notice that both fractions have similar denominators, \(x - 3\) and \(3 - x\). These can be related since \(3 - x\) is the negative of \(x - 3\).

Step 2: Rewrite the Denominators

Rewrite the expression \(\frac{9}{3-x}\) as \(-\frac{9}{x-3}\) to make the denominators identical. Now, the expression becomes: \(\frac{9}{x-3} + \left(-\frac{9}{x-3}\right)\).

Step 3: Combine the Fractions

Combine the fractions since they now have the same denominator: \(\frac{9 - 9}{x-3}\).

Step 4: Simplify the Expression

Calculate the numerator: \(9 - 9 = 0\). This simplifies the fraction to \(\frac{0}{x-3}\).

Step 5: Final Simplification

The fraction \(\frac{0}{x-3}\) is equal to 0, assuming \(x - 3 eq 0\).

Key Concepts

Fractions with Variables

Fractions with variables can often seem intimidating. However, understanding the basics of how they behave is the key to mastering them. Variables in fractions play the same role as numbers. They can be in the numerator, the denominator, or both. When fractions have variables:

• They can be combined just like numerical fractions, as long as they have the same denominator.
• The value of the variable can affect the fraction, sometimes making it undefined, like if the denominator is zero.

The exercise we are looking at deals with variables in the denominators. This means that we have to be careful about how we manipulate the expressions. It’s crucial to observe the denominators of the fractions to ensure they can be combined or if they’re already set to cancel each other out. Make sure the denominators are represented accurately and pay close attention to how they are structured. For instance, \(\frac{9}{x-3}\) and \(\frac{9}{3-x}\) are similar but not identical; the second can be rewritten for uniformity.

Combining Like Terms

Combining like terms is one of the fundamentals of algebra that helps in simplifying expressions. Like terms are terms that have the exact same variable part, making them combinable. The key steps involved in combining like terms are:

• Identifying terms with the same variable components.
• Combining them by adding or subtracting coefficients.

In our exercise, once the denominators are aligned to \(x-3\), \(\frac{9}{x-3} + \left(-\frac{9}{x-3}\right)\) means we combine the like terms in the numerators.The operation reduces to \(9 - 9\), which simplifies to zero. This step illustrates the power of recognizing and combining like terms, especially when they effectively nullify each other, resulting in a simplified form.

Algebraic Denominators

Handling algebraic denominators requires understanding expressions that involve variables in the denominator. The denominators dictate how we treat the entire fraction and how operations such as addition or subtraction can be performed:

• Ensure that denominators are equivalent before combining fractions.
• Observe how expressions change upon rewriting, like turning \(3-x\) into \(-1\times (x-3)\)\, which gives the same result with a clearer path to simplification.

To add or subtract fractions with algebraic denominators, a common denominator is essential. By rewriting \(\frac{9}{3-x}\) as \(-\frac{9}{x-3}\), the common denominator \(x-3\) emerges, allowing simple combination and eventual simplification.Once the fractions share the same denominator, it becomes straightforward to manage operations effectively, leading to results like \(\frac{0}{x-3}\), underlining the importance of handling such denominators accurately.