Problem 82 - Introductory Algebra for College Students

Question

After adding \(\frac{3 x+1}{4}\) and \(\frac{x+2}{4},\) I simplified the sum by dividing the numerator and the denominator by 4 I use similar procedures to find each of the following sums: $$ \frac{3}{8}+\frac{1}{8} \text { and } \frac{x}{x^{2}-1}+\frac{1}{x^{2}-1} $$

Step-by-Step Solution

Answer

The sum \( \frac{3}{8}+\frac{1}{8} \) equals \( \frac{1}{2} \) and the sum \( \frac{x}{x^{2}-1}+\frac{1}{x^{2}-1} \) equals \( \frac{x+1}{x^{2}-1} \).

Step 1: Add the Numerators

For the sum \( \frac{3}{8}+\frac{1}{8} \), simply add the numerators 3 and 1 to get 4. So, \( \frac{3}{8}+\frac{1}{8}=\frac{4}{8} \).

Step 2: Simplify the Result

Next, simplify the fraction \( \frac{4}{8} \) by dividing both the numerator and the denominator by 4 to get \( \frac{1}{2} \).

Step 3: Add the Numerators for the Second Sum

For the sum \( \frac{x}{x^{2}-1}+\frac{1}{x^{2}-1} \), do the same operation: add the numerators x and 1 to get \( x+1 \).

Step 4: Write the Final Result

So, \( \frac{x}{x^{2}-1}+\frac{1}{x^{2}-1}=\frac{x+1}{x^{2}-1} \). This fraction cannot be further simplified.

Key Concepts

Adding Fractions

When it comes to adding fractions, the process is straightforward if the denominators are the same. You keep the denominator and simply add the numerators. For instance, consider the sum \( \frac{3}{8} + \frac{1}{8} \). Since the denominators are both 8, you add the numerators 3 and 1 to obtain 4, leading to \( \frac{4}{8} \).

Solving a problem involving the addition of fractions with the same denominator involves adherence to this simple rule: conserve the denominator and accumulate the numerators. This approach allows the focus to be directed toward the numerators only, thereby simplifying the task at hand.

• Identify like denominators
• Add the numerators while maintaining the common denominator
• Simplify the result if possible

Before proceeding, ensure that the fractions are in their simplest form to facilitate easier operations in subsequent steps.

Simplifying Numerators

Once you've added the numerators, it's essential to see if you can simplify the numerators to make the overall expression smaller and more comprehensible. Upon obtaining \( \frac{4}{8} \) after adding the numerators, we should check for any common factors between the numerator and the denominator.

In the example above, both 4 and 8 can be divided by the number 4. By doing so, we reduce \( \frac{4}{8} \) to \( \frac{1}{2} \), which is the simplest form of the fraction. Simplifying fractions aids in understanding the magnitude of the number represented by the fraction and in performing further arithmetic operations.

• Look for common factors in both the numerator and the denominator
• Divide both the numerator and the denominator by the common factor
• Ensure the fraction is in its simplest form

This simplification step is crucial for reducing fractions to their lowest terms, which can be advantageous when comparing fractions or combining them with other terms.

Algebraic Expression Simplification

Moving beyond simple numerical examples, algebraic expression simplification requires similar principles but applies them to algebraic terms. Consider the sum \( \frac{x}{x^2-1} + \frac{1}{x^2-1} \). Here, the denominators are already the same, so we combine the numerators to obtain \( \frac{x+1}{x^2-1} \).

The next step is to check if this fraction can be simplified. In this particular case, the denominator is a difference of squares, \(x^2-1 = (x+1)(x-1)\), but the numerator isn't a multiple of the entire denominator expression, therefore it cannot be divided out.

As a final step, always inspect whether the expression can be factored further or if there are common factors that can be canceled out.

• Combine like terms in the numerator
• Attempt to factor both numerator and denominator
• Seek and cancel common factors if present

While the addition principle is the same as with numerical fractions, algebraic fractions require an additional eye for factoring and simplification to reach the most reduced form.