Problem 28 - Algebra 1

Question

Simplify the expression. $$\frac{x}{x^{2}+5 x-24}+\frac{8}{x^{2}+5 x-24}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{x + 8}{x^{2}+5x-24}\).

Step 1: Identify the Like Denominators

The two fractions \(\frac{x}{x^{2}+5x-24}\) and \(\frac{8}{x^{2}+5x-24}\) have the same denominator which is \(x^{2}+5x-24\).

Step 2: Combine Numerators

Having the same denominators, these two fractions can be combined into one by simply adding the numerators: \(x + 8\).

Step 3: Write the Final Simplified Expression

By combining the numerators, the simplified expression is \(\frac{x + 8}{x^{2}+5x-24}\).

Key Concepts

Understanding Like Denominators

When simplifying algebraic fractions, one crucial step is to work with 'like denominators.' Alike denominators occur when two or more fractions share the same bottom number or expression. This is important because it allows us to easily combine the fractions. Think of it like having slices of the same-sized pie; it's straightforward to see how many pies you have in total.

For example, let’s consider our fractions \(\frac{x}{x^{2}+5x-24}\) and \(\frac{8}{x^{2}+5x-24}\). These algebraic expressions have identical denominators, \(x^{2}+5x-24\). If the denominators had been different, we would have had to manipulate the expressions to find a common denominator. But since they're already the same, we're set to move forward to the next step, simplifying by combining numerators.

Combining Numerators

Once we've established like denominators in our algebraic fractions, as in our original expressions \(\frac{x}{x^{2}+5x-24}\) and \(\frac{8}{x^{2}+5x-24}\), we can proceed with 'combining numerators.' Combining numerators is simply the process of adding or subtracting the top numbers or expressions of fractions that have the same denominator.

Adding the Numerators

Our fractions have the numerators 'x' and '8'. If we add them together, we get \(x + 8\). Since the denominators are alike, we place this new numerator over the original common denominator, resulting in a single fraction. This process can be thought of as gathering terms to streamline the expression.

Simplifying Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. The goal in simplifying them is to condense the expression as much as possible, often making it easier to evaluate or utilize in further calculations. When we simplified the expression \(\frac{x}{x^{2}+5x-24}+\frac{8}{x^{2}+5x-24}\), we ended with \(\frac{x + 8}{x^{2}+5x-24}\), which is a single rational expression.

Further Simplification is Possible

It's worth mentioning that in some cases, these expressions can be simplified further by factoring. If the numerator and denominator have a common factor, it can be cancelled out. However, in our final simplified expression, factoring can be attempted, but we don't have enough information to proceed without knowing the factors of the denominator. Therefore, we consider \(\frac{x + 8}{x^{2}+5x-24}\) the simplest form of the original exercise.