Question

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

Short answer

The sum is \(\frac{x^2 + 8x - 9}{(x-3)(x+3)}\).

Step-by-step solution

Find a common denominator

To add the fractions \(\frac{x+1}{x-3}+\frac{4}{x+3}\), we need a common denominator. The common denominator of \(x-3\) and \(x+3\) is \((x-3)(x+3)\).

Rewrite each fraction with the common denominator

Rewrite each fraction with the common denominator \((x-3)(x+3)\). \(\frac{x+1}{x-3}\) becomes \(\frac{(x+1)(x+3)}{(x-3)(x+3)}\), and \(\frac{4}{x+3}\) becomes \(\frac{4(x-3)}{(x-3)(x+3)}\).

Combine the fractions

Add the fractions \(\frac{(x+1)(x+3)}{(x-3)(x+3)} + \frac{4(x-3)}{(x-3)(x+3)}\). Combine the numerators over the common denominator: \(\frac{(x+1)(x+3) + 4(x-3)}{(x-3)(x+3)}\).

Expand the numerators

Distribute and combine like terms in the numerators. \((x+1)(x+3) = x^2 + 3x + x + 3 = x^2 + 4x + 3\) and \(4(x-3) = 4x - 12\). Thus, the numerator becomes \(x^2 + 4x + 3 + 4x - 12\).

Simplify the numerator

Combine the like terms in the numerator: \(x^2 + 4x + 4x + 3 - 12 = x^2 + 8x - 9\). So, the combined fraction is \(\frac{x^2 + 8x - 9}{(x-3)(x+3)}\).

Key concepts

Common Denominator

When adding or subtracting rational expressions, finding a common denominator is essential. The common denominator is a shared multiple of the individual denominators. In this exercise, we need to add \(\frac{x+1}{x-3}\) and \(\frac{4}{x+3}\). To do this, we must identify a common denominator between \((x-3)\) and \((x+3)\), which is \((x-3)(x+3)\). This common denominator allows us to combine the fractions into a single expression over the same base.

Remember, having a common denominator simplifies the process of combining, because it converts the fractions to have the same base, making addition or subtraction straightforward. If denominators are not the same, converting them to a common one is a must.

Combining Fractions

Once we have a common denominator, we rewrite each fraction with this new common denominator. For instance: \(\frac{x+1}{x-3}\) is rewritten as \(\frac{(x+1)(x+3)}{(x-3)(x+3)}\) and \(\frac{4}{x+3}\) is rewritten as \(\frac{4(x-3)}{(x-3)(x+3)}\).

Now that both fractions have the common denominator \((x-3)(x+3)\), we combine them by adding their numerators:

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

Combine the fractions by combining the numerators over the common denominator: \(\frac{(x+1)(x+3) + 4(x-3)}{(x-3)(x+3)}\).

Simplifying Algebraic Expressions

The next step is to expand and simplify the numerator of the combined fraction. Using the distributive property:

• ull{ull}(x+1)(x+3) = x^2 + 3x + x + 3 = x^2 + 4x + 3
• ull{ull}4(x-3) = 4x - 12

Adding these results together:

\(x^2 + 4x + 3 + 4x - 12 = x^2 + 8x - 9\)

This gives us the simplified numerator: \(x^2 + 8x - 9\).

So the expression now reads: \(\frac{x^2 + 8x - 9}{(x-3)(x+3)}\).

Distributing and Combining Like Terms

In the final step, we ensure all terms are combined and simplified. Distribute the multiplicands and combine like terms in the numerator to achieve the simplest form.

Here's the step-by-step combining and simplifying: Start with the expanded terms from the numerator: \((x+1)(x+3)\) and \4(x-3)\.

• ull{ull}(x+1)(x+3) = x^2 + 4x + 3
• ull{ull}4(x-3) = 4x - 12

Then, combine like terms to simplify:
\x^2 + 4x + 3 + 4x - 12 = x^2 + 8x - 9\.

This leaves us with the final simplified fraction: \(\frac{x^2 + 8x - 9}{(x-3)(x+3)}\). This understanding of distributing and combining allows us to efficiently simplify complex expressions.