Verified Answer :

  • Analyze the denominators: \(3x+6 = 3(x+2)\) and \(x+2\).
  • Find the LCD: \(3(x+2)\).
  • Convert the second fraction to have the common denominator: \(\frac{x}{x+2} = \frac{3x}{3(x+2)}\).
  • Combine the fractions to get \(\frac{3x+11}{3x+6}\) for \(x eq -2\).
  • Final simplified result: \(\boxed{\frac{3x+11}{3x+6}}\).

Explanation

  1. Analysis of the Expression and Domain

The expression to simplify is \(\frac{11}{3x+6} + \frac{x}{x+2}\). First, we identify the domain of the expression. The denominators must not be zero:

  1. For \(3x+6\), \(3x+6 eq 0 \implies 3(x+2) eq 0 \implies x eq -2\).
  2. For \(x+2\), \(x+2 eq 0 \implies x eq -2\).

Thus, the expression is defined for all \(x eq -2\).

  1. Finding a Common Denominator

To add the fractions, they need a common denominator. We factor the first denominator:
\(3x+6 = 3(x+2)\).
The original expression becomes:
\[\frac{11}{3(x+2)} + \frac{x}{x+2}\]
The least common denominator (LCD) is \(3(x+2)\). We rewrite the second term by multiplying its numerator and denominator by 3:
\[\frac{x}{x+2} = \frac{x \cdot 3}{(x+2) \cdot 3} = \frac{3x}{3(x+2)}\]

  1. Simplifying the Sum

Now that both terms have the same denominator, we add their numerators:
\[\frac{11}{3(x+2)} + \frac{3x}{3(x+2)} = \frac{11 + 3x}{3(x+2)}\]
This expression is simplified, as there are no further common factors between the numerator and the denominator. Thus, the simplified expression is \(\frac{3x+11}{3x+6}\).

Examples

In engineering, simplifying algebraic expressions allows for easier circuit analysis or structural calculations. For example, when determining the equivalent resistance of resistors in parallel, you often end up with complex rational expressions. Simplifying these expressions helps identify key parameters like gain or frequency response more efficiently. Understanding how to combine and simplify these expressions is a foundational skill in applied mathematics and physics.

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