Question

Find the sum or difference. \(\frac{3}{x+4}-\frac{1}{x+6}\)

Short answer

The difference of the given fractions is \(\frac{2x+14}{(x+4)(x+6)}\)

Step-by-step solution

- Find the Common Denominator

The common denominator of the two fractions, \(\frac{3}{x+4}\) and \(\frac{1}{x+6}\), can be found by multiplying the two denominator expressions, giving us \((x+4)(x+6)\) as the common denominator.

- Rewrite the Fractions with the Common Denominator

Rewrite the fractions with the common denominator. The first fraction becomes \(\frac{3(x+6)}{(x+4)(x+6)}\) and the second fraction becomes \(\frac{1(x+4)}{(x+4)(x+6)}\).

- Perform the Subtraction

Now subtract the second fraction from the first one: \(\frac{3(x+6)}{(x+4)(x+6)} - \frac{1(x+4)}{(x+4)(x+6)}\).

- Simplify the Numerator

Expand and simplify the numerator: \(\frac{3x+18-x-4}{(x+4)(x+6)}\) leads to \(\frac{2x+14}{(x+4)(x+6)}\).

- Simplify Further if Possible

Look for the potential to simplify the result further. In this case, no further simplification is possible.

Key concepts

Common Denominator

When dealing with rational expressions, finding a common denominator is crucial for operations like addition and subtraction. This is because it creates a shared baseline that allows you to directly operate on the numerators. To find a common denominator, consider the denominators of both expressions involved.

For the exercise, the denominators are

• \(x + 4\)
• \(x + 6\)

The simplest way to find a common denominator is to multiply these two expressions, resulting in \((x+4)(x+6)\).

By forming this product, you effectively create a structure that can support both fractions. This allows them to "talk" in the same language, so to speak, setting the stage for the upcoming subtraction.

Fraction Subtraction

Once a common denominator is established, you're ready to perform subtraction. Fraction subtraction is similar to integer subtraction but focuses on the numerators since the denominators are now a match. Begin by rewriting each fraction so they both have the common denominator.

In this case,

• Rewrite the first fraction as \( \frac{3(x+6)}{(x+4)(x+6)} \)
• Rewrite the second fraction as \( \frac{1(x+4)}{(x+4)(x+6)}\)

Now, you subtract the second numerator from the first while keeping the common denominator intact:
\[\frac{3(x+6)}{(x+4)(x+6)} - \frac{1(x+4)}{(x+4)(x+6)}\]
Think of the denominator as a constant for this step, simplifying your task to merely dealing with what's above the line.

Expression Simplification

Expression simplification is the art of making math expressions as neat and tidy as possible. After subtracting the numerators, you’ll often have an expression that can be simplified further, which is the case here. The initial step involves expanding and combining like terms within the numerator.

• First, expand: \(3(x + 6) = 3x + 18\) and \(1(x + 4) = x + 4\).
• After the expansion, subtract: \(3x + 18 - (x + 4)\) results in \(3x + 18 - x - 4\).
• Combine like terms: \(3x - x = 2x\) and \(18 - 4 = 14\), giving the numerator \(2x + 14\).

Now, your fraction looks like:\[\frac{2x + 14}{(x+4)(x+6)}\]At this stage, always check for further simplification. Simplifying might involve factoring or reducing expressions, but in this case, \(2x + 14\) can't be factored in relation to the denominator, so this is your simplified result.