Problem 118 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \frac{\frac{1}{x-1}+\frac{2}{x}}{2-\frac{1}{x}} $$

Step-by-Step Solution

Answer

\( \frac{3x-2}{(x-1)(2x-1)} \)

Step 1: Find a Common Denominator for the Numerator

The numerator of the expression is \( \frac{1}{x-1} + \frac{2}{x} \). To add these fractions together, we need a common denominator. The denominators are \( x-1 \) and \( x \). A common denominator is their product, \( x(x-1) \).

Step 2: Rewrite the Numerator with Common Denominator

Rewriting each term of the numerator, we get:- \( \frac{1}{x-1} = \frac{x}{x(x-1)} \)- \( \frac{2}{x} = \frac{2(x-1)}{x(x-1)} \)So, the combined numerator becomes \( \frac{x + 2(x-1)}{x(x-1)} \).

Step 3: Simplify the Numerator

Inside the fraction, combine the terms:\[ x + 2(x-1) = x + 2x - 2 = 3x - 2 \]Thus, the numerator simplifies to \( \frac{3x-2}{x(x-1)} \).

Step 4: Simplify the Denominator

The denominator of the expression is \( 2 - \frac{1}{x} \). Rewrite this by finding a common denominator, which is \( x \):\[ 2 - \frac{1}{x} = \frac{2x}{x} - \frac{1}{x} = \frac{2x-1}{x} \]

Step 5: Write the Whole Expression

Substitute the simplified numerator and denominator back into the main fraction:\[ \frac{\frac{3x-2}{x(x-1)}}{\frac{2x-1}{x}} \]

Step 6: Simplify the Overall Expression

To simplify \( \frac{\frac{3x-2}{x(x-1)}}{\frac{2x-1}{x}} \), multiply by the reciprocal of the denominator:\[ \frac{3x-2}{x(x-1)} \cdot \frac{x}{2x-1} = \frac{x(3x-2)}{x(x-1)(2x-1)} \]

Step 7: Cancel Common Terms

Notice that \( x \) in the numerator and \( x \) in the first part of the denominator cancel out, leaving:\[ \frac{3x-2}{(x-1)(2x-1)} \]

Step 8: Present the Simplified Expression

The expression is now simplified to:\( \frac{3x-2}{(x-1)(2x-1)} \).

Key Concepts

Simplifying Fractions

When dealing with rational expressions, simplifying fractions is a crucial skill. Think of this as making complicated math look simpler and more understandable.

• To simplify a fraction, you want to reduce it to its simplest form.
• Always start by trying to factor both the numerator and denominator.
• Look for common factors that appear in both the top and bottom of the fraction. Often, you can "cancel out" these common factors.

In our exercise, once the overall fraction was constructed, we had the chance to cancel a common factor of \( x \), which made the expression less complex. Reducing expressions not just looks neat, but it also helps in performing further operations and understanding the relationships among variables.

Common Denominator

Finding a common denominator is like finding a common language between fractions to allow their addition or subtraction. When fractions have different denominators, you can't just directly add or subtract them.
Here's how you do it:

• Identify the denominators involved. In our case, these were \( x-1 \) and \( x \).
• Find the least common multiple (LCM) of these denominators. The LCM often involves multiplying them together, as seen with \( x(x-1) \).
• Rewrite each fraction with this common denominator. This might require multiplying the numerator and denominator by the necessary factors.

This way, the fractions become compatible, and you can combine them. This single denominator strategy maintains the balance in algebraic operations, allowing for seamless simplifications or incorporations of multiple terms.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (like addition or multiplication). They can represent a wide variety of things, from simple numbers to complex relationships.
In our problem, both the numerator and denominator of the rational expression included algebraic expressions that involved variables \( x \).

• Always check if these expressions can be broken down into simpler parts (factoring) or rearranged to make the math more manageable.
• Understanding the structure of these expressions helps in performing operations like addition, subtraction, or even simplification.
• In many exercises, simplifying algebraic expressions to combine or cancel terms is often the goal. For instance, combining the \( x + 2(x-1) = 3x-2 \) made it easier to simplify further.

Handling algebraic expressions intersects with key topics like factoring, expanding, and simplifying, which are foundational in algebra. Grasping these concepts allows you to manipulate expressions successfully within various mathematical problems.