Problem 37 - Elementary Algebra

Question

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{1}{x+3}+\frac{1}{x}}{\frac{1}{x+3}} $$

Step-by-Step Solution

Answer

Simplified result: \(2 + \frac{3}{x}\)

Step 1: Simplify the numerator

The numerator is \(\frac{1}{x+3} + \frac{1}{x}\). To add these fractions, find a common denominator which is \((x+3)x\). Thus, \(\frac{1}{x+3}\) becomes \(\frac{x}{x(x+3)}\), and \(\frac{1}{x}\) becomes \(\frac{x+3}{x(x+3)}\).

Step 2: Combine the fractions

Combine the fractions in the numerator: \(\frac{x}{x(x+3)} + \frac{x+3}{x(x+3)} = \frac{x + x + 3}{x(x+3)} = \frac{2x + 3}{x(x+3)}\). Now, the expression becomes \(\frac{\frac{2x+3}{x(x+3)}}{\frac{1}{x+3}}\).

Step 3: Simplify the complex fraction

To simplify the complex fraction \(\frac{\frac{2x+3}{x(x+3)}}{\frac{1}{x+3}}\), multiply the numerator by the reciprocal of the denominator: \(\frac{2x+3}{x(x+3)} \times \frac{x+3}{1}\). The \((x+3)\) terms cancel out, leaving \(\frac{2x+3}{x}\).

Step 4: Simplify the result

The remaining expression is \(\frac{2x+3}{x}\). This can be split into two separate fractions: \(\frac{2x}{x} + \frac{3}{x} = 2 + \frac{3}{x} \).

Key Concepts

Common Denominator

When adding or subtracting fractions, a common denominator is essential. This step ensures both fractions share the same base, making it easy to combine them.
For instance, in the given problem, we look at \(\frac{1}{x+3}\) and \(\frac{1}{x}\).
The denominators are \(x+3\) and \(x\), so the common denominator will be \(x(x+3)\).

Thus, to rewrite each fraction over the common denominator:
\[\frac{1}{x+3} = \frac{x}{x(x+3)}\]
and
\[\frac{1}{x} = \frac{x+3}{x(x+3)}\]
It's like finding a common stage for them to perform together! Understanding this makes it easier to move forward.

Combine Fractions

Once both fractions share a common denominator, combining them is straightforward.
In our example, we combine \(\frac{x}{x(x+3)}\) and \(\frac{x+3}{x(x+3)}\):
\[\frac{x}{x(x+3)} + \frac{x+3}{x(x+3)} = \frac{x + (x + 3)}{x(x+3)}\]
Notice that we simply add the numerators while keeping the common denominator intact.
Combining gives us: \[\frac{2x + 3}{x(x+3)}\]

Fractions with the same base make this process seamless, turning multiple elements into one unified piece.

Simplify Complex Fraction

Simplifying a complex fraction means resolving a fraction made up of other fractions.
Looking at our problem: \[\frac{\frac{2x+3}{x(x+3)}}{\frac{1}{x+3}}\]
We do this by multiplying the numerator by the reciprocal of the denominator:
\[\frac{2x+3}{x(x+3)} \times \frac{x+3}{1}\]
The \(x+3\) in the numerator and denominator cancel each other out:
\[\frac{2x + 3}{x}\]

This step drastically simplifies complex expressions by removing unnecessary components, leading to a more straightforward result.

Algebraic Fractions

Algebraic fractions contain variables in their numerators, denominators, or both. Simplifying them follows the same principles as numerical fractions but requires handling algebraic expressions.
With our final result: \[\frac{2x + 3}{x}\]
We split it into two separate fractions to further simplify:
\[\frac{2x}{x} + \frac{3}{x}= 2 + \frac{3}{x}\]

Tip: Always look for common factors and opportunities to cancel terms.
Understanding these steps becomes easier with practice, turning seemingly complex problems into simple, solvable puzzles.