Question

Subtract \(\frac{3 x}{2 x-10}-\frac{x}{2 x+6}\). Write your answer in lowest terms. A. \(\frac{4 x^2+28 x}{(2 x-10)(2 x+6)}\) B. \(\frac{x^2+14 x}{(x-5)(x+3)}\) C. \(\frac{2 x^2+14 x}{2(x-5)(x+3)}\) D. \(\frac{x(x+7)}{(x-5)(x+3)}\)

Short answer

Option A: \(\frac{4x^2 + 28x}{(2x-10)(2x+6)}\).

Step-by-step solution

Find the Least Common Denominator (LCD)

Identify the least common denominator for the fractions. The denominators are \(2x - 10\) and \(2x + 6\). The LCD is the product of these denominators: \ \((2x - 10)(2x + 6)\).

Rewrite Fractions with the LCD

Rewrite each fraction with the common denominator: \ \(\frac{3x}{2x-10} = \frac{3x(2x+6)}{(2x-10)(2x+6)}\) \ \(\frac{x}{2x+6} = \frac{x(2x-10)}{(2x-10)(2x+6)}\).

Simplify the Numerators

Expand and simplify the numerators: \ \(3x(2x+6) = 6x^2 + 18x\) \ \(x(2x-10) = 2x^2 - 10x\).

Subtract the Numerators

Subtract the second numerator from the first: \ \(6x^2 + 18x - (2x^2 - 10x)\) \ Simplify the expression: \ \(6x^2 + 18x - 2x^2 + 10x = 4x^2 + 28x\).

Write in Lowest Terms

Write the result in lowest terms: \ \(\frac{4x^2 + 28x}{(2x-10)(2x+6)}\).

Check the Answer Choices

Compare the simplified result with the provided choices. The answer is given by option A: \ \(\frac{4x^2 + 28x}{(2x-10)(2x+6)}\).

Key concepts

Subtracting Fractions

When subtracting fractions, it's important to have the same denominator for both fractions. This common denominator allows you to directly subtract the numerators. Remember:

• Ensure both fractions share the same denominator.
• Combine the numerators by subtraction while keeping the shared denominator.
• Simplify the resulting fraction if possible.

In this exercise, we need to subtract \(\frac{3x}{2x-10}\) and \(\frac{x}{2x+6}\). Before doing so, we must ensure they have the same denominator.

Least Common Denominator

The least common denominator (LCD) is the smallest multiple that is common to the denominators of two or more fractions. To subtract fractions, find the LCD and rewrite each fraction with this denominator.

In our problem, the denominators are \(2x-10\) and \(2x+6\). So:

• Factor each denominator, \(2x-10 = 2(x-5)\) and \(2x+6 = 2(x+3)\).
• The LCD is then the product of these factored forms, \(2(x-5) \times 2(x+3) = 4(x-5)(x+3)\).
• Rewrite each fraction using this LCD.

That gives us: \(\frac{3x(2x+6)}{(2x-10)(2x+6)}\) and \(\frac{x(2x-10)}{(2x-10)(2x+6)}\).

Simplifying Expressions

After finding the LCD and rewriting each fraction, the next step is simplifying the numerators. Simplifying involves expanding the numerators' expressions and then combining like terms.

For this exercise:

• First, expand the numerators: \(3x(2x+6) = 6x^2 + 18x\) and \(x(2x-10) = 2x^2 - 10x\).
• Then combine the numerators by subtraction: \(6x^2 + 18x - (2x^2 - 10x)\).
• Simplify the resulting expression: \(6x^2 + 18x - 2x^2 + 10x = 4x^2 + 28x\).

The simplified numerator is now \(4x^2 + 28x\).

Factoring Polynomials

Factoring polynomials means expressing a polynomial as the product of its factors. This step is crucial for simplifying the final expression.

To factor the resulting polynomial from our exercise:

• Identify the greatest common factor (GCF) in the numerator \(4x^2 + 28x\).
• The GCF is \(4x\).
• So, factor this out: \(4x(x + 7)\).
• Our fractions had the common denominator \((2x-10)(2x+6)\) which we factored as: \(4(x-5)(x+3)\).

This step ensures we correctly simplify the whole expression to \(\frac{4x(x+7)}{4(x-5)(x+3)} = \frac{x(x+7)}{(x-5)(x+3)}\).