Problem 43 - Prealgebra and Introductory Algebra

Question

Simplify. $$\frac{x+4}{2 x}-\frac{x-1}{x^{2}}$$

Step-by-Step Solution

Answer

\( \frac{x^2+2x +2}{2x^2} \)

Step 1: Common Denominator

For subtracting the fractions, transform both fractions to have the same denominator, yielding this equation:\( \frac{x+4}{2x} \cdot \frac{x}{x} - \frac{x-1}{x^2} \cdot \frac{2}{2} \)This simplifies to: \( \frac{x(x+4)}{2x^2} - \frac{2(x-1)}{2x^2} \)

Step 2: Subtracting Fractions

Now that both fractions have the same denominator, subtract the numerators to simplify the equation:\( \frac{x(x+4) - 2(x-1)}{2x^2} \)which simplifies to: \( \frac{x^2+4x - 2x +2}{2x^2} \)

Step 3: Simplify The Equation

Now simplifying the numerator:\( \frac{x^2+2x +2}{2x^2} \)

Key Concepts

Common Denominator

When dealing with fractions, it's imperative that they share the same denominator before you add or subtract them. This step is known as finding a common denominator. For example, consider the fractions \( \frac{x+4}{2x} \) and \( \frac{x-1}{x^2} \). These fractions have different denominators (\(2x\) and \(x^2\), respectively). To proceed with subtraction, we need to transform them to have a uniform denominator.
To find this common denominator, we consider the least common multiple (LCM) of both denominators. In our case, the LCM is \(2x^2\). This means both fractions will need to be adjusted so that their denominators read \(2x^2\).

• Multiply the first fraction by \(\frac{x}{x}\) to convert the denominator to \(2x^2\).
• Multiply the second fraction by \(\frac{2}{2}\) to attain the same denominator.

The fractions then transform to \( \frac{x(x+4)}{2x^2} \) and \( \frac{2(x-1)}{2x^2} \). This harmonization is crucial, as it simplifies the next steps in the process.

Subtracting Fractions

With a common denominator established, the next step is to subtract the fractions. This involves focusing on their numerators since the denominators will now remain constant. For the fractions \( \frac{x(x+4)}{2x^2} \) and \( \frac{2(x-1)}{2x^2} \), subtraction can proceed smoothly because their denominators are identical.
To subtract, simply take the first numerator and subtract the second from it:

• First numerator: \( x(x+4) \)
• Second numerator: \( 2(x-1) \)

The operation becomes: \[ x(x+4) - 2(x-1) \]Simplifying within the numerator provides: \[ x^2 + 4x - 2x + 2 \]This reduction leads to a more manageable problem and sets the stage for the final step, simplifying the resulting expression further.

Polynomial Simplification

Polynomial simplification is the process used to refine a polynomial expression to its simplest form. After subtracting fractions, you often face such a task. In our case, the expression derived from the subtraction step is \( \frac{x^2 + 2x + 2}{2x^2} \).
The goal here is to reduce the expression to its most condensed form. Observe the numerator: \( x^2 + 2x + 2 \).

• Check for common factors in the terms. Here, there are no common factors across all terms.
• Look for other simplifications, such as factoring or canceling common terms with the denominator. However, make sure the numerator and denominator terms share a common factor that can be simplified. In this case, they do not.

When there are no further simplifications, the expression remains as is, presented in its simplest form. This final check ensures you've achieved the thorough simplification necessary for a neat and accurate result.