Problem 60 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

The simplified expression is \( \frac{-1}{(x+h)x} \).

Step 1: Identify the Expression Structure

The given expression is a fraction, where the numerator is a difference of two fractions and the denominator is a single variable term, 'h'. The expression is \( \frac{\frac{1}{x+h} - \frac{1}{x}}{h} \).

Step 2: Find a Common Denominator for the Numerator

We need to simplify the numerator, which is \( \frac{1}{x+h} - \frac{1}{x} \). To do this, find the common denominator for the two fractions, which is \( (x+h)x \). Rewrite each fraction with this common denominator:\[\frac{1}{x+h} = \frac{x}{(x+h)x}, \quad \frac{1}{x} = \frac{x+h}{(x+h)x}\]

Step 3: Simplify the Numerator

Subtract the fractions in the numerator:\[\frac{x}{(x+h)x} - \frac{x+h}{(x+h)x} = \frac{x - (x + h)}{(x+h)x} = \frac{x - x - h}{(x+h)x} = \frac{-h}{(x+h)x}\]

Step 4: Simplify the Entire Expression

Now substitute back the simplified numerator into the original expression:\[\frac{\frac{-h}{(x+h)x}}{h} = \frac{-h}{(x+h)x} \times \frac{1}{h}\]This simplifies to:\[\frac{-h}{(x+h)x} \cdot \frac{1}{h} = \frac{-1}{(x+h)x}\]

Step 5: State the Final Simplified Expression

The expression is fully simplified to:\[\frac{-1}{(x+h)x}\]

Key Concepts

Common Denominator

In the realm of algebraic fractions, finding a common denominator is a crucial step for adding or subtracting different fractions. This process is foundational because it lets us combine the fractions into a single expression.

When dealing with fractions such as \( \frac{1}{x+h} \text{ and } \frac{1}{x} \), it's important to find a common term that both denominators can divide.
The common denominator here is the product of the two original denominators: \( (x+h)x \).

This choice ensures that each fraction can be rewritten in terms of this shared base, allowing for seamless subtraction or addition. To convert each fraction, multiply both its numerator and denominator by whatever factor it needs to match the common denominator.

For example:

• \( \frac{1}{x+h} \) becomes \( \frac{x}{(x+h)x} \)
• \( \frac{1}{x} \) becomes \( \frac{x+h}{(x+h)x} \)

With these transformations, the fractions are ready for further operations.

Fraction Subtraction

Subtracting fractions involves uniting them under a common denominator, and then combining their numerators accordingly. Once you have a common denominator, subtract the numerators, treating them as a single fraction.

For instance, using our expression, after finding the common denominator \((x+h)x\), the subtraction process would be:

• Numerator 1: \( \frac{x}{(x+h)x} \)
• Numerator 2: \( \frac{x+h}{(x+h)x} \)

Combine these by subtracting their numerators:\[\frac{x}{(x+h)x} - \frac{x+h}{(x+h)x} = \frac{x - (x+h)}{(x+h)x}\]

This step yields a new numerator \( x - (x + h) \), simplifying to \( -h \). With the common divisor in place, the numerators seamlessly combine, resulting in a new, simpler fraction.

Expression Simplification

Simplifying an expression involves reducing it to its most basic form without changing its value. This often means combining like terms and reducing fractions.

In our context, once the fraction subtraction gives us \( \frac{-h}{(x+h)x} \), align both the numerator and denominator to cancel out common factors if possible.

In this particular example, since there's an 'h' in both the numerator and a factor outside the fractional operand (which is the 'h' in the denominator), you can simplify it further by cancelling 'h':

• The negative sign remains in the result.
• The resulting expression, \( \frac{-1}{(x+h)x} \), is the simplified form.

This process is a clear demonstration of expression simplification.

Algebraic Fractions

Algebraic fractions are similar to regular fractions, but they can contain variables in the numerators, the denominators, or in both. Understanding these fractions is essential because they frequently appear in algebra.

With variables included, operations require a solid grasp on algebraic rules, especially when the goal is to simplify or equate them.

For example, in the expression \( \frac{\frac{1}{x+h} - \frac{1}{x}}{h} \), you are tasked with understanding how to manipulate these algebraic fractions to reach a simplified outcome.

The steps taken:

• Find a common denominator.
• Perform operations such as subtraction.
• Simplify the resulting fraction.

Each step requires careful handling of both numerical and variable components to ensure accuracy. Recognizing patterns and applying strategies for dealing with variables is key to mastering algebraic fractions.