Problem 33 - Elementary and Intermediate Algebra

Question

Perform the operations. Simplify, if possible. $$ \frac{3 x}{2 x-1}-\frac{2 x}{2 x+3} $$

Step-by-Step Solution

Answer

\( \frac{2x^2 + 11x}{(2x-1)(2x+3)} \)

Step 1: finding a common denominator

To subtract fractions, we first need a common denominator. For the fractions \( \frac{3x}{2x-1} \) and \( \frac{2x}{2x+3} \), the common denominator is the product of the two denominators: \((2x-1)(2x+3)\).

Step 2: Rewrite each fraction

Rewrite each fraction with the common denominator. For the first fraction: \( \frac{3x}{2x-1} \) becomes \( \frac{3x(2x+3)}{(2x-1)(2x+3)} \). For the second fraction: \( \frac{2x}{2x+3} \) becomes \( \frac{2x(2x-1)}{(2x-1)(2x+3)} \).

Step 3: Subtract the two fractions

Subtract the fractions: \[ \frac{3x(2x+3)}{(2x-1)(2x+3)} - \frac{2x(2x-1)}{(2x-1)(2x+3)} = \frac{3x(2x+3) - 2x(2x-1)}{(2x-1)(2x+3)} \].

Step 4: Simplify the numerator

Simplify the expression in the numerator: \[ 3x(2x+3) - 2x(2x-1) = 6x^2 + 9x - 4x^2 + 2x = 2x^2 + 11x \].

Step 5: Write the final expression

The expression with the simplified numerator becomes: \[ \frac{2x^2 + 11x}{(2x-1)(2x+3)} \].We check if further simplification is possible, but it is not in this case.

Key Concepts

Common Denominator

When dealing with algebraic fractions, especially in subtraction, finding a common denominator is crucial. It allows us to combine the fractions into one expression, making operations like addition and subtraction possible. For fractions such as \( \frac{3x}{2x-1} \) and \( \frac{2x}{2x+3} \), neither denominator is a factor of the other. Therefore, we multiply the two denominators to create a common one. The product \((2x-1)(2x+3)\) becomes the common denominator. This approach might seem cumbersome at first, but it systematically resolves the inequality of denominators. It ensures that each fraction is expressed in terms of the same base, enabling direct operations on the numerators.

Fraction Subtraction

Once both fractions have a common denominator, subtraction becomes straightforward. By aligning the fractions over the same denominator, we easily subtract their numerators:

• Place the first fraction’s adjusted numerator, like \(3x(2x+3)\), over the common denominator.
• For the second fraction, adjust similarly: \(2x(2x-1)\) over the common denominator.
• Subtract the numerators directly to find the new numerator: \(3x(2x+3) - 2x(2x-1)\).

This operation only affects the numerators and doesn’t change the denominator. Always ensure your subtraction is accurate by carefully aligning each term.

Simplifying Expressions

Simplifying expressions begins with the numerator. After subtracting, you often face a more complex expression. Here, \(3x(2x+3) - 2x(2x-1)\) transforms through expansion:

• Expand using the distributive property: for \(3x(2x+3)\), multiply to get \(6x^2 + 9x\); for \(-2x(2x-1)\), ensure to distribute the negative, giving \(-4x^2 + 2x\).
• Combine like terms to condense the expression: \(6x^2 + 9x - 4x^2 + 2x\) simplifies to \(2x^2 + 11x\).

This step checks if any factors cancel with the denominator, which simplifies the fraction further. Often, simplifying reveals whether further reduction is achievable, streamlining complex results to their simplest form. Here, no further simplification is possible.