Problem 68 - Elementary Algebra

Question

Add or subtract as indicated and express your answers in simplest form. (Objective 3) $$\frac{4}{x-8}+\frac{9}{x}$$

Step-by-Step Solution

Answer

\( \frac{13x - 72}{x(x-8)} \)

Step 1: Identify the Least Common Denominator (LCD)

To add the fractions \( \frac{4}{x-8} \) and \( \frac{9}{x} \), we need a common denominator. The denominators are \( x-8 \) and \( x \). Therefore, the least common denominator (LCD) is \( x(x-8) \).

Step 2: Rewrite Each Fraction with the LCD

Rewrite each fraction so that they both have the LCD of \( x(x-8) \) as their denominator. Multiply the numerator and denominator of \( \frac{4}{x-8} \) by \( x \), and \( \frac{9}{x} \) by \( x-8 \). This gives:\( \frac{4x}{x(x-8)} + \frac{9(x-8)}{x(x-8)} \).

Step 3: Combine the Fractions

Now that both fractions have the same denominator, combine them into a single fraction:\( \frac{4x + 9(x-8)}{x(x-8)} \).

Step 4: Simplify the Numerator

Distribute the 9 in the expression \( 9(x-8) \) which gives \( 9x - 72 \). Add the terms in the numerator:\( 4x + 9x - 72 = 13x - 72 \).

Step 5: Write the Final Expression in Simplest Form

The expression is now \( \frac{13x - 72}{x(x-8)} \). This is in its simplest form, as no common factors exist between the numerator and the denominator.

Key Concepts

least common denominator

When working with fractions, especially in algebra, finding a common ground is crucial for simplification. This is where the concept of the **least common denominator (LCD)** becomes relevant. The LCD is the smallest expression that both denominators can divide into without leaving a remainder. In our exercise, the denominators are \( x-8 \) and \( x \). The LCD for these expressions is determined by multiplying them together, resulting in \( x(x-8) \).

• Multiply both denominators together if they are distinct.
• If denominators share a variable or term, ensure it is only counted once.

Using the LCD ensures that when we perform operations like addition or subtraction, all fractions involved are speaking the same 'language'. This translation into a common denominator is critical for the operations to proceed smoothly, just like converting currencies before comparing or adding them.

fraction addition

Adding fractions in algebra isn't that different from adding regular fractions. The primary task is to ensure that the fractions have the same denominator before combining them. Once the least common denominator (LCD) is set, the fractions are rewritten to reflect this common base. For the fractions \( \frac{4}{x-8} \) and \( \frac{9}{x} \), the LCD is \( x(x-8) \).

• Multiply the numerator and the denominator of each fraction by the necessary factor to reach the LCD.
• This keeps the value of each fraction the same.
• Our resulting fractions become \( \frac{4x}{x(x-8)} \) and \( \frac{9(x-8)}{x(x-8)} \).

After securing a common denominator, the next step is simple. We combine the numerators: adding them together and placing the result over the common denominator. This process is similar to combining like terms in algebraic expressions and ensures the operation is mathematically sound.

simplification of fractions

Simplifying fractions in the realm of algebra means reducing them to their simplest form where possible. This process can involve several steps, which often include simplifying the numerator and ensuring there are no common factors between the numerator and the denominator.

• Distribute any coefficients or terms within the expression. For example, distribute the 9 in \( 9(x–8) \) to get \( 9x - 72 \).
• Combine like terms within the numerator. In our exercise: \( 4x + 9x - 72 = 13x - 72 \).
• Check for any common factors that can be divided out, though in our case, there are none.

The aim here is to ensure that the expression is as reduced as possible without losing accuracy. Simplified forms are easier to interpret and use in subsequent operations. It's akin to tidying up a room; removing unnecessary clutter to see the essentials clearly. By simplifying \( \frac{13x - 72}{x(x-8)} \), we ensure that the result is both elegant and concise, ready for any further analysis or use.