Problem 16 - Intermediate Algebra

Question

Complete each solution. $$\frac{x^{2}+3 x}{x-1}-\frac{2 x-1}{x-1}=\frac{x^{2}+3 x-(}{x-1}$$ $$=\frac{x^{2}+3 x-2 x\square 1}{x-1}$$ $$=\frac{x^{2}+\square+\square}{x-1}$$

Step-by-Step Solution

Answer

Simplified expression: \( \frac{x^{2} + x + 1}{x-1} \).

Step 1: Understand the Problem

The problem involves simplifying the expression \( \frac{x^{2}+3x}{x-1} - \frac{2x-1}{x-1} \). Both fractions have the same denominator \( x-1 \), so we can combine them into a single fraction.

Step 2: Combine the Fractions

Since both fractions have the same denominator, we subtract the numerators directly: \( (x^{2}+3x) - (2x-1) \). This gives us: \[ \frac{x^{2}+3x - (2x-1)}{x-1} \].

Step 3: Simplify the Numerator

Distribute the negative sign in the subtraction: \( x^{2} + 3x - 2x + 1 \). This simplifies further to \( x^{2} + x + 1 \).

Step 4: Final Expression

Now, substitute the simplified numerator back into the fraction: \[ \frac{x^{2} + x + 1}{x-1} \]. This is the simplified form of the original expression.

Key Concepts

Combining Fractions

Combining fractions is a fundamental aspect of working with rational expressions. When dealing with algebraic fractions (or rational expressions), the goal is often to combine them into a single fraction to simplify the expression.

Let's explore how to do this by focusing on the denominators: for two fractions to be combined, they must have the same denominator. This means looking at expressions like \( \frac{x^2 + 3x}{x-1} \) and \( \frac{2x - 1}{x-1} \). Here, both fractions share a common denominator of \( x-1 \).

This shared denominator allows us to

• Subtract the numerators directly.
• Combine them into a single expression: \( x^2 + 3x - (2x - 1) \).

Once you have a single fraction, it becomes easier to manipulate and simplify further. Combining fractions is an invaluable technique, especially in algebra, where complex expressions are the norm.

Simplifying Expressions

Simplifying expressions is a crucial part of making algebra more manageable. After combining fractions, the next logical step is to simplify the expression to its most concise form. This involves several key ideas:

1. **Distributing Negative Signs:** When you have an expression like \((x^2 + 3x) - (2x - 1)\), the negative sign must be distributed across the terms in the parenthesis. This means transforming it to \(x^2 + 3x - 2x + 1\).

2. **Combining Like Terms:** In the expression \(x^2 + 3x - 2x + 1\), notice that \(3x\) and \(-2x\) are like terms. Combining them leads to \(x^2 + x + 1\). This process removes redundant terms and simplifies the expression structure.

Simplification streamlines algebraic expressions, making them faster and easier to work with in future calculations.

Algebraic Manipulation

Algebraic manipulation is an essential skill in solving equations and simplifying expressions. It encompasses a range of techniques used to rewrite expressions in different forms.

In this exercise, we already saw algebraic manipulation through

• Distributing negative signs.
• Combining like terms.

These actions transformed \(x^2 + 3x - (2x - 1)\) into \(x^2 + x + 1\).

Other methods of algebraic manipulation include:

• **Factoring:** Breaking down an expression into simpler multiplied factors
• **Expanding:** Spreading factors into a series of summed or subtracted terms

Algebraic manipulation is critical for solving equations because it allows you to change how an equation looks without changing its solutions.