Problem 57 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$\frac{\frac{3}{x-1}-\frac{3}{a-1}}{x-a}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{-3}{(x-1)(a-1)} \).

Step 1: Identify a common denominator

To simplify the expression \( \frac{\frac{3}{x-1}-\frac{3}{a-1}}{x-a} \), let's first focus on the numerator \( \frac{3}{x-1} - \frac{3}{a-1} \). We need a common denominator to combine these fractions. In this case, the common denominator is \((x-1)(a-1)\).

Step 2: Rewrite fractions with a common denominator

Rewrite each fraction with the common denominator: The first fraction becomes \( \frac{3(a-1)}{(x-1)(a-1)} \) and the second fraction becomes \( \frac{3(x-1)}{(x-1)(a-1)} \).

Step 3: Combine the fractions in the numerator

Now subtract the second fraction from the first: \[ \frac{3(a-1) - 3(x-1)}{(x-1)(a-1)} = \frac{3a - 3 - 3x + 3}{(x-1)(a-1)} \]Simplify the expression: \( \frac{3a - 3x}{(x-1)(a-1)} \).

Step 4: Simplify the expression by factoring

Factor out a 3 from the numerator: \( \frac{3(a-x)}{(x-1)(a-1)} \).

Step 5: Divide the simplified numerator by \(x-a\)

Now, return to the original problem and divide by \(x-a\): \[ \frac{\frac{3(a-x)}{(x-1)(a-1)}}{x-a} = \frac{3(a-x)}{(x-1)(a-1) (x-a)} \].Since \(a-x = -(x-a)\), we simplify to: \( \frac{-3}{(x-1)(a-1)} \).

Key Concepts

Common Denominator

When you work with fractions, finding a common denominator is often necessary to perform operations like addition or subtraction.
To simplify the given expression, we start by addressing the numerator part: \( \frac{3}{x-1} - \frac{3}{a-1} \). The key here is to find a single denominator that both fractions can agree on.

• Each original denominator, \(x-1\) and \(a-1\), is different, so we need something that incorporates both.
• The simplest way to achieve this is by multiplying these denominators together, resulting in \((x-1)(a-1)\).
• This combination becomes our common denominator, allowing us to rewrite each fraction equivalently so they can be directly subtracted.

Finding a common denominator helps ensure that all parts of the fraction refer to the same whole, which is essential for accurate subtraction and further simplification.

Factoring Expressions

Factoring is another crucial step in simplifying expressions, especially when dealing with polynomials. Once we've brought everything under a common denominator, we can use factoring to further break down or simplify the components.

• In our numerator, \( 3a - 3x\), we notice that both terms have a common factor of 3.
• By factoring out this common factor, the expression becomes \( 3(a-x) \).
• This step not only simplifies the expression but also makes it clearer and easier to work with in subsequent operations.

Factoring simplifies expressions by reducing them to a product of simpler expressions. This can often reveal underlying patterns or cancellations, making the arithmetic or algebraic manipulation more straightforward.

Subtracting Fractions

Subtracting fractions can be a bit tricky, particularly if they have different denominators. As previously explained, we first achieve this by finding a common denominator.
Here’s a simple guide for subtracting fractions:

• Once fractions share the same denominator, their numerators can be directly subtracted.
• For the given problem, subtract the numerators: \(3(a-1) - 3(x-1)\).
• Simplify the result to \(3a - 3x\) by distributing the multiplication.

After subtraction, combine like terms if possible, and always check if further simplification is achievable.
In our example, simplifying gave us the opportunity to factor the numerator. These steps ensure your final expression is in its simplest, most workable form, which is an essential skill in algebra.