Problem 50 - Algebra and Trigonometry

Question

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 a Common Denominator

To subtract the fractions, we need a common denominator. The denominators here are \(x-4\) and \(x+6\). The common denominator will be their product: \((x-4)(x+6)\).

Step 2: Adjust the Fractions

Rewrite each fraction with the common denominator \((x-4)(x+6)\). Multiply the numerator and denominator of \(\frac{x}{x-4}\) by \(x+6\), and \(\frac{3}{x+6}\) by \(x-4\) to get: \[\frac{x(x+6)}{(x-4)(x+6)} - \frac{3(x-4)}{(x-4)(x+6)}.\]

Step 3: Expand the Numerators

Expand the expressions in the numerator:\[x(x+6) = x^2 + 6x\text{ and }3(x-4) = 3x - 12.\]

Step 4: Perform the Subtraction

Subtract the second expanded numerator from the first:\[\frac{x^2 + 6x - (3x - 12)}{(x-4)(x+6)} = \frac{x^2 + 6x - 3x + 12}{(x-4)(x+6)}.\]

Step 5: Simplify the Expression

Combine like terms in the numerator:\[x^2 + 3x + 12.\]Thus, the expression becomes \[\frac{x^2 + 3x + 12}{(x-4)(x+6)}.\] This is the simplified form of the given expression.

Key Concepts

Common Denominator

When dealing with fractions, whether adding or subtracting, having a common denominator is crucial. The denominator is essentially the bottom part of a fraction. For example, in the fraction \( \frac{a}{b} \), \( b \) is the denominator. It determines how many parts the fraction represents of the whole.
In algebraic fractions, denominators can be expressions like \( x-4 \) or \( x+6 \). To add or subtract fractions with different denominators, we need to find a common denominator that both terms can share. This means both fractions must have the same bottom number or expression.

• Step 1: Identify the different denominators.
• Step 2: Determine the least common denominator (LCD) by finding a value or expression that contains each denominator as a factor.
• Step 3: For the exercise, this was done by multiplying \( x-4 \) and \( x+6 \) to get the common denominator \((x-4)(x+6)\).

Now each fraction can be rewritten to have this common denominator, setting the stage for subtraction.

Subtraction of Fractions

Once fractions have a common denominator, the process of subtraction becomes straightforward. For the exercise provided, after establishing the common denominator \((x-4)(x+6)\), each fraction needs to be rewritten using this denominator.
Here's how to subtraction of algebraic fractions works:

• Multiply the numerator and the denominator of one fraction to make its denominator match with the common denominator.
• Repeat this for the other fraction.
• Subtract the numerators while keeping the common denominator the same.

Remember, the goal is to subtract the numerators of the fractions since the denominators are now aligned. For example, if your fractions are \( \frac{a}{d} - \frac{b}{d} \), they become \( \frac{a-b}{d} \) after subtracting where \(d\) is our common denominator.

Simplifying Expressions

The final step in manipulating algebraic fractions is simplifying the expression. This means reducing it to its simplest form where no further operations can simplify it further.
Here's how simplification unfolds:

• First, perform any expansions needed on the numerators or denominators. For our exercise, we expanded \( x(x+6) = x^2 + 6x \) and \( 3(x-4) = 3x - 12 \).
• Next, combine like terms. In the numerator, this means adding or subtracting coefficients of the same variable power, such as reducing \( x^2 + 6x - 3x + 12 \) to \( x^2 + 3x + 12 \).
• Finally, look for any common factors in the numerator and denominator that can be divided out. This will give the simplest version of your algebraic fraction.

Simplifying expressions not only makes the solution more understandable but is also essential for solving equations efficiently. In the end, it could help in reducing errors in your calculations.