Problem 23 - Elementary Algebra

Question

For Problems 1-40, perform the indicated operations and express answers in simplest form. $$ \frac{x-1}{5 x+5}-\frac{x-4}{3 x+3} $$

Step-by-Step Solution

Answer

The simplified form is \(\frac{-2x + 17}{15x + 15}\).

Step 1: Identify the Common Denominator

To subtract fractions, we first need a common denominator. The denominators in the given expression are \(5x + 5\) and \(3x + 3\). Notice that both denominators can be factored: \(5x + 5\) can be factored as \(5(x + 1)\) and \(3x + 3\) as \(3(x + 1)\). The common denominator is \(15(x + 1)\), which is the least common multiple of the two denominators.

Step 2: Rewrite Each Fraction with the Common Denominator

We rewrite each fraction with the common denominator \(15(x + 1)\).1. Multiply the first fraction by \(3/3\): \[ \frac{x-1}{5(x+1)} \cdot \frac{3}{3} = \frac{3(x-1)}{15(x+1)} \]2. Multiply the second fraction by \(5/5\): \[ \frac{x-4}{3(x+1)} \cdot \frac{5}{5} = \frac{5(x-4)}{15(x+1)} \]

Step 3: Subtract the Fractions

Subtract the numerators of the fractions:\[ \frac{3(x-1) - 5(x-4)}{15(x+1)} \]Distribute within the numerators:\[ 3(x-1) = 3x - 3 \]\[ 5(x-4) = 5x - 20 \]Subtract:\[ (3x - 3) - (5x - 20) = 3x - 3 - 5x + 20 \]

Step 4: Simplify the Numerator

Combine like terms in the numerator:\[ 3x - 5x + 20 - 3 = -2x + 17 \]Thus, the resulting expression becomes:\[ \frac{-2x + 17}{15(x+1)} \]

Step 5: Verify the Simplest Form

Check if \(-2x + 17\) and \(15(x+1)\) have common factors. Since they do not, the fraction is already in simplest form. Therefore, the simplified expression is:\[ \frac{-2x + 17}{15x + 15} \]

Key Concepts

Subtracting Rational Expressions

When working with algebraic fractions, subtracting rational expressions involves a few vital steps.
The most crucial step is ensuring that both fractions are working with a common denominator.
Behind this process, we imagine a scenario quite similar to subtracting simple numerical fractions like \(\frac{1}{2} - \frac{1}{3}\). The key process involves converting both fractions to share the same base, in this case, the denominator.

• Identify the denominators of each fraction. In our example, they were \(5x + 5\) and \(3x + 3\).
• Factor each denominator if possible, making subtracting easier later.
• Calculate the least common denominator; this step ensures consistency across all fractions involved.
• Once the common denominator is set, rewrite each fraction using it.
• Finally, align the numerators over this shared base and subtract them directly.

Subtracting numerators once they have a common base ensures a smooth transition to simplifying the expression.

Factoring Polynomials

When factoring polynomials, you convert expressions into simpler products.
If you remember factoring from earlier math classes, it was about breaking down numbers; the same idea applies, but with variables involved.
Let's break it down:

• Identify parts that are common in each term of the polynomial, often seen in their coefficients.
• In our exercise, this led us to express \(5x + 5\) as \(5(x + 1)\) and \(3x + 3\) as \(3(x + 1)\).
• Factoring like this helps in matching the denominators when subtracting fractions. It also makes it easier to spot the least common multiple.

Remember, the goal is to rewrite the expression in terms of its basic elements, making operations like subtraction and simplification much less tedious.

Least Common Multiple

Finding the least common multiple (LCM) for denominators is essential in smoothing the subtraction of rational expressions.
The LCM represents the smallest number or expression that can be evenly divided by both original denominators.
Here's how we achieved it here:

• After factoring, see what terms the denominators share. In this case, both had \((x+1)\) as a factor.
• Multiply each unique factor the greatest number of times it occurs in each original denominator.
• This process gave us \(15(x+1)\) as the common denominator.
• Using the least common multiple allows us to restructure each fraction, allowing direct arithmetic operations on the numerators.

This step is crucial as it paves the way for clean and efficient subtraction, making simplifying expressions that much easier at the end.

Simplifying Expressions

Simplifying expressions is the final piece of the puzzle after subtracting rational expressions.
Once the numerators have been subtracted, simplify further to ensure the expression is in its simplest form.
Let's walk through it:

• Start by combining like terms in the numerator. In our example, we maneuvered \(3x - 5x + 20 - 3\) into \(-2x + 17\).
• Check your new expression against the common denominator for common factors that can be divided out.
• While no common factors between the numerator and denominator meant the simplification was complete here, if there were, you would strip them out.

Simplifying is often a satisfying end to the task, stripping an expression to its clean core form and confirming its accuracy by verifying you have reduced it as much as possible.