Problem 58 - College Algebra and Calculus: An Applied Approach

Question

Perform the indicated operations and simplify.\(\frac{4}{x+2}-6\)

Step-by-Step Solution

Answer

The simplified form of the given equation \( \frac{4}{x+2} - 6 \) is \( \frac{-2(3x+4)}{x+2} \)

Step 1: Rewrite

Rewrite \(-6\) as \(-6 \times \frac{x+2}{x+2}\) giving us \( \frac{4}{x+2} - \frac{6(x+2)}{x+2} \)

Step 2: Perform Operations

Subtract the two fractions, as both denominators are the same, which results in \( \frac{4 - 6(x+2)}{x+2} \)

Step 3: Simplify the Numerator

Expand and simplify the numerator, we have \( \frac{4 - 6x -12}{x+2} \) which reduces further to \( \frac{-6x -8}{x+2} \)

Step 4: Simplify by factoring

Simplify by factoring, if possible. The top 'numerator' can be factored into \( -2(3x + 4) \)

Step 5: Final Simplification

Our final simplification is \( \frac{-2(3x + 4)}{x+2} \)

Key Concepts

Operations with Fractions

Understanding operations with fractions is vital when dealing with algebraic expressions. When fractions have the same denominator, their operations are straightforward. If we wish to subtract one fraction from another, like in the exercise \(\frac{4}{x+2}-6\), we need to ensure they have a common denominator before we can perform the subtraction.

For instance, to subtract \(6\) from \(\frac{4}{x+2}\), we convert \(6\) into a fraction with the same denominator: \(\frac{6(x+2)}{x+2}\). This conversion allows us to perform the subtraction directly as shown in the steps of the problem solving process. The numerators of the fractions are then combined, following the arithmetic operation, while the denominator remains unchanged.

Simplifying Expressions

The core of simplifying expressions lies in making them easier to understand and solve. Simplification can involve combining like terms, reducing fractions, or eliminating unnecessary complexity. In the given example, once we have a common denominator, we simplify by combining the numerators.

For our exercise, simplification occurs when expanding \(4 - 6(x+2)\) to \(4 - 6x - 12\) and then combining the constants (4 and -12) to get the simpler form \(\frac{-6x - 8}{x+2}\). This step is crucial as it takes the expression to its most reduced form, resulting in a clearer and more concise expression ready for further analysis or operations.

Factoring Algebraic Expressions

Factoring is essential to algebraic expressions, as it often reveals simpler forms and can make other operations, such as simplifying fractions, easier. To factor an algebraic expression, look for common factors in each term.

In our exercise, \( -6x - 8\) can be factored by finding the greatest common factor (GCF), which in this case is \( -2\). Extracting \( -2\) from each term in the numerator gives us \( -2(3x + 4)\). Recognizing common factors and factoring allows further reduction when possible. However, for our specific problem, the expression \(\frac{-2(3x + 4)}{x+2}\) cannot be simplified further by factoring since there are no common factors in the numerator and the denominator. Factoring has still provided us with a cleaner, more manageable form of the expression.