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Chapter 11: Problem 9

Simplify. $$\frac{3 x}{2 x+3}+\frac{5 x}{2 x+3}$$

Short Answer

Expert verified
\(\frac{8x}{2x + 3}.\)

Step by step solution

01

Identify Common Denominator

The terms \(\frac{3x}{2x + 3}\) and \(\frac{5x}{2x + 3}\) have the same denominator, which is \(2x + 3\).
02

Combine Terms

Since the denominators are the same, the numerators can be combined. This leads to \(\frac{3x + 5x}{2x + 3}\).
03

Simplify Numerator

The numerator can be simplified to \(8x\), so the simplified expression is \(\frac{8x}{2x + 3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Denominator
When dealing with rational expressions, a common denominator is essential for combining or comparing fractions. A rational expression is a fraction where both the numerator and the denominator are polynomials. In rational expressions, just like in ordinary fractions, the denominators play a crucial role. To perform addition or subtraction, the expressions must share a common denominator.

In the exercise given, both terms share a denominator of \(2x + 3\). This makes the process of simplifying much easier, as the challenge of finding or adjusting denominators is already solved. Some key points to remember include:
  • Only numerators need to be combined when denominators are identical.
  • Ensure that all expressions are simplified as much as possible before proceeding with any operations.
Combining Like Terms
Combining like terms involves merging terms that have the same variables raised to the same power. It is a fundamental algebraic principle used to simplify and solve equations or expressions. Think of it as 'cleaning up' an expression to make complex problems more manageable.

The original expression \(\frac{3x}{2x + 3} + \frac{5x}{2x + 3}\) allows for this combination because they have like terms in the numerator: \(3x\) and \(5x\). Here鈥檚 how combining happens:
  • First, confirm that terms are indeed like terms - they must share the same variable and exponent.
  • Add their coefficients together, which involves simple arithmetic: \(3 + 5 = 8\).
  • Place the new combined term into the numerator: \(\frac{8x}{2x + 3}\).

By doing this, you arrive at a much simpler expression, making further calculations easy to handle.
Numerator Simplification
Numerator simplification is a necessary step once numerators have been combined. It involves making the numerator as simple and sleek as possible, often by combining or canceling terms. This process ensures clarity and prepares the expression for potential further operations.

In our exercise, after identifying that the numerators \(3x\) and \(5x\) can be added, we get \(8x\), resulting in \(\frac{8x}{2x + 3}\). This new expression is cleaner and carries no unnecessary complexity. Here are the steps:
  • Ensure the combined numerator is correct: check your arithmetic.
  • Simplify the expression if possible, by factoring even further in different scenarios.
  • Final check that the expression cannot be reduced any further.

This expression \(\frac{8x}{2x + 3}\) is now simplified, making it much easier to understand and work with in future problems.

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