Problem 40 - Algebra and Trigonometry

Question

\(35-54\) . Perform the addition or subtraction and simplify. $$ \frac{x}{x-4}-\frac{3}{x+6} $$

Step-by-Step Solution

Answer

\( \frac{x^2 + 3x + 12}{(x-4)(x+6)} \)

Step 1: Identify the Problem

We need to subtract two fractions: \( \frac{x}{x-4} \) and \( \frac{3}{x+6} \). Before subtracting, we must have a common denominator.

Step 2: Find a Common Denominator

The denominators are \( x-4 \) and \( x+6 \). The least common denominator (LCD) is their product: \((x-4)(x+6)\).

Step 3: Adjust the Numerators

Rewrite each fraction with the common denominator. Multiply \( \frac{x}{x-4} \) by \( \frac{x+6}{x+6} \) and \( \frac{3}{x+6} \) by \( \frac{x-4}{x-4} \). - First fraction: \( \frac{x(x+6)}{(x-4)(x+6)} = \frac{x^2 + 6x}{(x-4)(x+6)} \).- Second fraction: \( \frac{3(x-4)}{(x-4)(x+6)} = \frac{3x - 12}{(x-4)(x+6)} \).

Step 4: Perform the Subtraction

Subtract the adjusted numerators: \( \frac{x^2 + 6x}{(x-4)(x+6)} - \frac{3x - 12}{(x-4)(x+6)} \).This becomes \( \frac{x^2 + 6x - 3x + 12}{(x-4)(x+6)} \).

Step 5: Simplify the Result

Combine like terms in the numerator:\( x^2 + 6x - 3x + 12 = x^2 + 3x + 12 \).So, the simplified result is \( \frac{x^2 + 3x + 12}{(x-4)(x+6)} \).

Key Concepts

Common Denominator

When dealing with fractions, whether numerical or algebraic, having a common denominator is crucial for performing operations like addition and subtraction. A common denominator is essentially the same denominator for multiple fractions, which allows us to directly combine them.

In the realm of algebraic fractions, finding a common denominator requires taking into account the algebraic expressions in the denominators. To find this 'common ground', we need to determine the least common denominator (LCD). This is typically the least common multiple (LCM) of the given denominators.

In the exercise above, we have the denominators \(x-4\) and \(x+6\). Their least common denominator is their product, \((x-4)(x+6)\). Once we have this common denominator, we can rewrite each fraction so that they both share this denominator. This makes the subtraction possible.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, making sure the expression is as concise as possible. This is an important step in solving or performing any operation on expressions, as it makes them easier to work with and understand.

After finding a common denominator and adjusting the fractions accordingly, which in this case are \(\frac{x(x+6)}{(x-4)(x+6)}\) and \(\frac{3(x-4)}{(x-4)(x+6)}\), the next critical step is to subtract the numerators.

It's essential to combine like terms at this stage. For instance, \(x^2 + 6x - 3x + 12\) simplifies to \(x^2 + 3x + 12\). This streamlined expression is much neater and allows for more natural integration into subsequent mathematical operations or final solutions.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, denominator, or both contain algebraic expressions (like polynomials). These kinds of fractions can be somewhat tricky at first, but they follow the same principles as regular numerical fractions.

In our scenario, the fractions \(\frac{x}{x-4}\) and \(\frac{3}{x+6}\) were involved. Operations with algebraic fractions typically require working through several steps. This includes ensuring denominators are equal, manipulating expressions to have the same bases, and simplifying the resulting expressions.

Mastering algebraic fractions opens up a robust field of problem-solving skills. As you become familiar with manipulating these expressions, you'll find yourself handling more complex algebraic equations with confidence. Remember, practice makes perfect! Always review each step carefully to ensure accuracy across your fractions.