Problem 19 - Elementary Algebra

Question

Simplify \(8+\frac{3}{x-6}\)

Step-by-Step Solution

Answer

Question: Simplify the expression \(8 + \frac{3}{(x-6)}\). Answer: \(\frac{8x - 45}{(x-6)}\)

Step 1: Determine the common denominator

The current denominator of the fraction is \((x-6)\). Since \(8\) can be written as a fraction with a denominator of \((x-6)\), the common denominator will be \((x-6)\).

Step 2: Rewrite the terms with the common denominator

Before combining the terms, we need to rewrite 8 with the common denominator, \((x-6)\). To do this, we will multiply the numerator and the denominator of 8 by the denominator, \((x-6)\): \(8 = \frac{8}{1}\) (rewrite \(8\) as a fraction) \(8 = \frac{8(x-6)}{(x-6)}\) (multiply the numerator and the denominator of 8 by \((x-6)\)) Now, the two terms have the same denominator: \(\frac{8(x-6)}{(x-6)} + \frac{3}{(x-6)}\)

Step 3: Combine the terms

Now that both terms have the same denominator, \((x-6)\), we can combine the numerators: \(\frac{8(x-6) + 3}{(x-6)}\)

Step 4: Distribute and simplify the numerator

Distribute the \(8\) in the numerator: \(\frac{8x - 48 + 3}{(x-6)}\) Combine the constant terms: \(\frac{8x - 45}{(x-6)}\)

Step 5: Present the simplified result

The simplified expression is: \(\frac{8x - 45}{(x-6)}\)

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Just like fractions, rational expressions can be simplified, combined, or compared. Here are some key ideas to keep in mind when working with them:

• The expression remains valid as long as the denominator doesn't become zero, since division by zero is undefined.
• The rules for adding, subtracting, multiplying, and dividing rational expressions are similar to those for regular fractions.
• It's important to identify the restrictions on the variable(s) to ensure the denominator is never zero in any term you end up with.

In the given expression, the rational part is \(\frac{3}{x-6}\). Our goal is to either simplify or perform operations without breaching the rule concerning division by zero.

Common Denominators

When adding or subtracting rational expressions, having a common denominator is crucial. This means both terms must have the same bottom number or expression.

• This allows us to easily add or subtract the fractions by merely combining the numerators, while keeping the common denominator unchanged.
• The process of finding a common denominator often involves multiplying terms so that their denominators match.

In the example, \(8\) (a whole number) is transformed into a rational expression with the denominator \((x-6)\), by writing it as \(\frac{8(x-6)}{x-6}\). This aligns it with \(\frac{3}{x-6}\), simplifying the process of combining the terms.

Simplifying Expressions

Simplifying an expression involves rewriting it in its simplest form. This often includes distributing, combining like terms, and reducing fractions. Let's break down these processes:

• Distribution: Apply the distributive property \(a(b+c) = ab + ac\), to expand expressions.
• Combining like terms: Bring together terms that have the same variable raised to the same power.
• Reduction: Cancel out any common numerical factors, either in the terms or in the entire expression.

In our example, after rewriting to a common denominator, we combine the numerators \(8(x-6) + 3\). Then, distribute to get \(8x - 48 + 3\) and simplify it to \(8x - 45\). The final simplified expression forms as \(\frac{8x - 45}{x-6}\). Understanding and executing these steps will help achieve a streamlined result for any rational expression.