Problem 82 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{x}{2 x-1}+\frac{1-x}{3 x} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{x^2 + 3x - 1}{(2x-1)(3x)} \).

Step 1: Find the Common Denominator

To add the fractions \( \frac{x}{2x-1} \) and \( \frac{1-x}{3x} \), we first need a common denominator. The denominators are \( 2x-1 \) and \( 3x \). The least common denominator (LCD) is \((2x-1)(3x)\).

Step 2: Rewrite Each Fraction with the LCD

Rewrite each fraction so they both have the denominator \((2x-1)(3x)\). Multiply the numerator and the denominator of the first fraction by \(3x\) to get \( \frac{3x^2}{(2x-1)(3x)} \). Multiply the numerator and the denominator of the second fraction by \(2x-1\) to get \( \frac{(1-x)(2x-1)}{(3x)(2x-1)} \).

Step 3: Expand the Numerator of the Second Fraction

Expand \((1-x)(2x-1)\) to express the second term clearly. This gives: \((1-x) \times (2x-1) = 2x - 1 - 2x^2 + x = -2x^2 + 3x - 1\).

Step 4: Add the Fractions

Combine the fractions using the common denominator \((2x-1)(3x)\):\[\frac{3x^2}{(2x-1)(3x)} + \frac{-2x^2 + 3x - 1}{(2x-1)(3x)} = \frac{3x^2 + (-2x^2 + 3x - 1)}{(2x-1)(3x)}.\]

Step 5: Simplify the Numerator

Simplify the expression in the numerator: \[3x^2 - 2x^2 + 3x - 1 = x^2 + 3x - 1.\]

Step 6: Write the Final Simplified Expression

The simplified expression is:\[\frac{x^2 + 3x - 1}{(2x-1)(3x)}.\] This is the simplified form of the original expression.

Key Concepts

Common Denominator

When adding or subtracting fractions, having a common denominator is crucial. Think of the denominator as a shared base or floor for both fractions. Without it, you can’t easily combine the fractions into one. To find a common denominator, you need to identify the least common multiple of the denominators of the fractions you are dealing with. This is essentially the smallest expression that both denominators can divide into without leaving a remainder.

In our exercise, the given denominators are \(2x-1\) and \(3x\). The least common denominator (LCD) of these is \((2x-1)(3x)\). By multiplying the denominators together, we create a single, common ground from which to work, allowing us to rewrite each fraction with this shared base.

Fraction Addition

Once a common denominator has been found and applied, the next step is to perform the fraction addition. This process involves rewriting each fraction to reflect the common denominator, allowing for proper addition of the numerators.

For the first fraction, \(\frac{x}{2x-1}\), you multiply both the numerator and the denominator by the missing part of the common denominator, which is \(3x\). This gives us \(\frac{3x^2}{(2x-1)(3x)}\). The second fraction \(\frac{1-x}{3x}\) is then rewritten by multiplying numerator and denominator by \(2x-1\), resulting in \(\frac{(1-x)(2x-1)}{(3x)(2x-1)}\).

With both fractions now having the common denominator, they can be written as a single fraction by adding their numerators together. This is presented as:\[\frac{3x^2}{(2x-1)(3x)} + \frac{-2x^2 + 3x - 1}{(2x-1)(3x)} = \frac{3x^2 + (-2x^2 + 3x - 1)}{(2x-1)(3x)}.\]

Polynomial Simplification

Polynomial simplification involves combining like terms within a polynomial, reducing it to its simplest form. This process is critical in making the final expression clearer and more manageable.

After adding the two fractions, we arrive at a single numerator: \(3x^2 + (-2x^2 + 3x - 1)\). To simplify, you combine like terms. The terms \(3x^2\) and \(-2x^2\) can be added to give \(x^2\), while the \(+3x\) and \(-1\) remain unchanged. This results in the simplified numerator: \(x^2 + 3x - 1\).

Thus, the final simplified expression of the original algebraic fraction addition problem is:\[\frac{x^2 + 3x - 1}{(2x-1)(3x)}.\]
The simplification not only symbolizes cleanliness in mathematics but also aids in solving further algebra problems by providing an understandable and concise expression.