Problem 31 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$\frac{2}{4 x-5}-\frac{5}{10 x+1}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{27}{(4x - 5)(10x + 1)}\).

Step 1: Identify a Common Denominator

To subtract fractions, we need to have a common denominator. In this expression, the denominators are \(4x - 5\) and \(10x + 1\). We multiply them together to get the least common denominator: \((4x - 5)(10x + 1)\).

Step 2: Rewrite Each Fraction with the Common Denominator

We convert each fraction to have a denominator of \((4x - 5)(10x + 1)\). For the first fraction \(\frac{2}{4x - 5}\), multiply both the numerator and denominator by \((10x + 1)\), resulting in \(\frac{2(10x + 1)}{(4x - 5)(10x + 1)}\). For the second fraction \(\frac{5}{10x + 1}\), multiply both the numerator and denominator by \((4x - 5)\), resulting in \(\frac{5(4x - 5)}{(4x - 5)(10x + 1)}\).

Step 3: Subtract the Fractions by Subtracting Numerators

Now that the fractions have a common denominator, subtract the numerators, resulting in:\[\frac{2(10x + 1) - 5(4x - 5)}{(4x - 5)(10x + 1)}\]

Step 4: Simplify the Numerator

Expand the terms in the numerator:\(2(10x + 1) = 20x + 2\)\(5(4x - 5) = 20x - 25\)Subtract these expressions: \(20x + 2 - (20x - 25) = 20x + 2 - 20x + 25 = 27\).

Step 5: Write Final Simplified Expression

The simplified form of our expression is:\[\frac{27}{(4x - 5)(10x + 1)}\]

Key Concepts

Common Denominator

In algebra, when dealing with fractions, a common denominator is essential for performing operations like addition or subtraction. The common denominator is a shared multiple of the denominators of each fraction in the expression.
This allows us to rewrite each fraction in terms of this common denominator, enabling straightforward calculation.

• For the expression \(\frac{2}{4x-5} - \frac{5}{10x+1}\), the denominators are different: \(4x-5\) and \(10x+1\).
• We find a common denominator by multiplying these individual denominators: \((4x - 5)(10x + 1)\).

This product becomes the least common denominator, a term often encountered in polynomial expressions. Using a common denominator simplifies further operations and ensures that all fractions in the expression "speak the same language." This way, you can simplify or compute the expressions without confusion.

Simplifying Expressions

Simplifying expressions is a core part of algebra and involves reducing an expression to its simplest form. This means making it easier to work with by performing basic operations and removing unnecessary complexity.

• In our example, once the fractions have been rewritten with a common denominator, the expression \(\frac{2(10x + 1)}{(4x - 5)(10x + 1)} - \frac{5(4x - 5)}{(4x - 5)(10x + 1)}\) is simplified step-by-step.
• This includes expanding, combining like terms, and ensuring there are no common factors left unchecked.

The result of simplifying our specific expression is the fraction \(\frac{27}{(4x - 5)(10x + 1)}\). Here, all possible reductions in the expression have been made, resulting in a simplified version that is easy to interpret and evaluate.

Subtraction of Fractions

Subtraction of fractions requires attentiveness to detail since it involves ensuring both parts of the equations use the same denominators. Without a common denominator, meaningful subtraction isn't possible.
After establishing a common denominator for our expression, we focus on the numerators.

• For the given algebraic fractions, we subtract the numerators: \(2(10x + 1)\) and \(5(4x - 5)\), while keeping the common denominator \((4x - 5)(10x + 1)\).
• The numerators themselves may require simplification; in our step-by-step solution, \(2(10x + 1)\) becomes \(20x + 2\), and \(5(4x - 5)\) becomes \(20x - 25\).

The subtraction process, \(20x + 2 - (20x - 25)\), is straightforward once the expressions are expanded and aligned. This leads us to cancel common terms, further streamlining the expression for the final simplified result of \(\frac{27}{(4x - 5)(10x + 1)}\).

Polynomial Expressions

Polynomial expressions are made up of variables and coefficients, involving terms that can be added, subtracted, or multiplied. These are fundamental in algebra and form the basis of more complex calculations.
In our exercise, we handle polynomial denominators \(4x - 5\) and \(10x + 1\). Understanding how to manipulate these forms the foundation for finding the common denominator.

• This includes expanding polynomials, which means to multiply out expressions fully. It requires taking each term of one polynomial and multiplying it through every term in the other polynomial.
• Also, recognizing that when both fractions share this expanded form, numerous algebraic operations, like subtraction, become substantially easier to perform.

Dealing with polynomial expressions is crucial, as it builds competency in algebra that is essential for tackling real-world mathematical problems and crossing into other math territories.