Question

Perform the indicated operation and simplify the result. Leave your answer in factored form. $$ \frac{7}{x-3}-\frac{3}{x+1} $$

Short answer

\( \frac{4(x+4)}{(x-3)(x+1)} \)

Step-by-step solution

Find a Common Denominator

To add or subtract fractions, a common denominator is needed. The denominators in this problem are \( x-3 \) and \( x+1 \). The common denominator will be the product of these two expressions: \( (x-3)(x+1) \).

Rewrite Each Fraction

Rewrite each fraction with the common denominator \( (x-3)(x+1) \). \( \frac{7}{x-3} = \frac{7(x+1)}{(x-3)(x+1)} \) and \( \frac{3}{x+1} = \frac{3(x-3)}{(x-3)(x+1)} \).

Combine the Fractions

Combine the two fractions: \( \frac{7(x+1)}{(x-3)(x+1)} - \frac{3(x-3)}{(x-3)(x+1)} \).

Subtract the Numerators

Subtract the numerators while keeping the common denominator: \( \frac{7(x+1) - 3(x-3)}{(x-3)(x+1)} \).

Simplify the Numerator

Expand and combine like terms in the numerator: \( 7(x+1) - 3(x-3) = 7x + 7 - 3x + 9 = 4x + 16 \). So the expression becomes: \( \frac{4x + 16}{(x-3)(x+1)} \).

Factor the Numerator

Factor the numerator if possible: \( 4x + 16 = 4(x + 4) \). The simplified expression is then: \( \frac{4(x + 4)}{(x-3)(x+1)} \).

Key concepts

Common Denominator

When dealing with algebraic fractions, finding a common denominator is crucial for performing addition or subtraction. The common denominator is essentially a common base that both denominators can convert to. This way, the fractions can seamlessly combine into one.
In our exercise, the denominators were \(x - 3\) and \(x + 1\). To find a common denominator, you multiply these two: \((x-3)(x+1)\). This allows the fractions to reframe with a shared base.

Factoring

Factoring plays a fundamental role in simplifying algebraic expressions. It involves breaking down an algebraic expression into simpler components or factors that, when multiplied together, reproduce the original expression.
In our solution, notice how we needed the numerator in factored form to simplify the final fraction expression. The original numerator was \(4x + 16\), and by factoring, we found that it equals \(4(x + 4)\). Catching this allows for a streamlined, simplified answer.

Simplifying Fractions

Simplifying fractions entails minimizing them to their simplest form, ensuring they are easier to work with and understand. This means reducing all components—numerators and denominators alike—whenever possible.
In our case study, we see this as crucial in the final step. The numerator \(4x + 16\) was simplified by factoring to \(4(x + 4)\). This makes the resultant fraction cleaner and easier to interpret: \(\frac{4(x + 4)}{(x-3)(x+1)}\).

Subtraction of Fractions

Subtracting algebraic fractions follows a methodical process, requiring a shared common denominator for any subtraction or addition to take place. In this exercise, upon converting our fractions to a common denominator \((x-3)(x+1)\), we could then focus on the numerators.
The subtraction process involved combining the numerators: \(7(x+1) - 3(x-3)\). Carefully carrying out this subtraction before simplifying the resulting numeric expression is key. The outcome was a cleaner numerator, facilitating simpler final steps in factoring and simplification.