Problem 119 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the addition or subtraction and simplify. $$\frac{1}{x+5}+\frac{x}{x-8}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \(\frac{x^2 + 6x - 8}{(x-8)(x+5)}\).

Step 1: Find the Common Denominator

Since the denominators of the two fractions are dissimilar and not constants, their common denominator will be the product of the denominators. Hence the common denominator in this case is \((x + 5)(x - 8)\).

Step 2: Rewrite the Fractions Using the Common Denominator

The first fraction will become \(\frac{1 * (x-8)}{(x+5)(x-8)}\) and the second fraction will become \(\frac{x * (x+5)}{(x-8)(x+5)}\). So the expression is now \(\frac{(x-8)}{(x+5)(x-8)} + \frac{x * (x+5)}{(x-8)(x+5)}\).

Step 3: Simplify the Expression

After simplifying, the expression becomes \(\frac{x - 8 + x^2 + 5x}{(x-8)(x+5)}\). This simplifies further to \(\frac{x^2 + 6x - 8}{(x-8)(x+5)}\).

Key Concepts

Common Denominator

When dealing with algebraic fractions, one key step in the addition process is identifying a common denominator. A common denominator is essentially a shared base for both fractions, enabling you to combine them into a single expression. When the denominators differ, as they do in algebraic fractions, the common denominator is often the product of both denominators. In our example, the denominators are \(x+5\) and \(x-8\). Their common denominator is \((x+5)(x-8)\). This ensures each fraction shares the same base and can be combined together.

Finding the common denominator might initially seem complex, but it's a systematic process. By multiplying both denominators together, you create a unified foundation to handle both fractions. This makes subsequent steps, like addition, far more straightforward.

Simplifying Expressions

Once you have successfully rewritten the fractions with a common denominator, the next important step is simplifying the resulting expression. Simplification involves combining like terms and making the expression as concise as possible, without altering its value. Initially, each term in the added fractions is expanded and organized in the numerator: \(x - 8 + x^2 + 5x\).

By collecting similar terms, this becomes \(x^2 + 6x - 8\). Simplification is crucial, as it provides a cleaner and more manageable form of the expression, which is important for clarity and further analysis. Keep in mind that simplifying does not change the overall value; it only refines how the expression is presented.

Addition of Fractions

Adding fractions, whether numerical or algebraic, requires careful alignment of denominators. In algebra, after aligning denominators by finding a common one, the numerators are combined. Consider our simplified example with the common denominator \((x+5)(x-8)\). The transformed expression becomes \(\frac{(x-8) + (x^2 + 5x)}{(x+5)(x-8)}\).

• Add the numerators: The coefficients of \(x\) and constant terms are combined.
• Result: You get a new fraction with a combined numerator \(x^2 + 6x - 8\).

The ability to perform this operation reliably allows you to tackle more complicated expressions and build on foundational skills that are essential as you progress in algebra.