Problem 60 - Algebra for College Students

Question

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{3}{2 x+1}+\frac{2}{3 x+4} $$

Step-by-Step Solution

Answer

The sum is \(\frac{13x+14}{(2x+1)(3x+4)}\).

Step 1: Find the Least Common Denominator (LCD)

The first step is to determine the least common denominator (LCD) of the fractions. The denominators are \(2x+1\) and \(3x+4\). Since these expressions are linear and do not have common factors, the LCD is simply their product: \[(2x+1)(3x+4)\]

Step 2: Rewrite Fractions with the LCD

Rewrite each fraction with the LCD as the new denominator. This requires modifying the numerators accordingly:For the first fraction: \[\frac{3}{2x+1} = \frac{3(3x+4)}{(2x+1)(3x+4)} = \frac{9x+12}{(2x+1)(3x+4)}\]For the second fraction:\[\frac{2}{3x+4} = \frac{2(2x+1)}{(2x+1)(3x+4)} = \frac{4x+2}{(2x+1)(3x+4)}\]

Step 3: Add the Numerators

Now add the numerators of the modified fractions while keeping the common denominator:\[\frac{9x+12}{(2x+1)(3x+4)} + \frac{4x+2}{(2x+1)(3x+4)} = \frac{(9x+12) + (4x+2)}{(2x+1)(3x+4)}\]Combine like terms in the numerator:\[\frac{13x+14}{(2x+1)(3x+4)}\]

Step 4: Simplify the Expression

Finally, check if the expression can be simplified. The numerator is \(13x+14\), and the denominator is \((2x+1)(3x+4)\). Since they don't have any common factors, the expression is already in its simplest form:\[\frac{13x+14}{(2x+1)(3x+4)}\]

Key Concepts

Least Common Denominator

In mathematics, when dealing with fractions, the least common denominator (LCD) is essential for tasks such as adding or subtracting fractions. The LCD represents the smallest shared multiple of the denominators of the fractions involved. By achieving a common denominator, we can easily combine the fractions.

• For rational expressions like \( \frac{3}{2x+1} \) and \( \frac{2}{3x+4} \), the denominators are linear polynomials.
• To find the LCD, we analyze the denominators \(2x+1\) and \(3x+4\), which do not share any common factors.
• Therefore, the LCD is the product of these two linear expressions: \((2x+1)(3x+4)\).

Determining the LCD is crucial as it ensures that the denominators are equivalent, which is necessary to perform addition or subtraction of the fractions. With the LCD set, we can proceed to manipulate each fraction to have this common denominator while adjusting their numerators accordingly.

Adding Fractions

Once the least common denominator is established, the next step is to adjust each fraction so they both have this LCD. This allows us to add the fractions seamlessly.

• We transform \( \frac{3}{2x+1} \) so that it has the LCD \((2x+1)(3x+4)\) by multiplying the numerator \(3\) by the expression \(3x+4\), resulting in \(9x + 12\).
• Similarly, \( \frac{2}{3x+4} \) becomes \( \frac{4x+2}{(2x+1)(3x+4)} \) by multiplying the numerator \(2\) by \(2x+1\).

Now, both fractions \( \frac{9x+12}{(2x+1)(3x+4)} \) and \( \frac{4x+2}{(2x+1)(3x+4)} \) share the common denominator \((2x+1)(3x+4)\).
This arrangement allows us to easily add their numerators. We simply combine them to get \(13x + 14\) while keeping the same denominator.
The results in the expression \( \frac{13x+14}{(2x+1)(3x+4)} \) which shows how the addition process works using a unified denominator.

Simplifying Expressions

The final step after the mathematical operation of adding fractions is simplification. Simplifying ensures that the expression is presented in the most efficient and concise manner possible.

• After adding the fractions, we arrive at \( \frac{13x + 14}{(2x+1)(3x+4)} \).
• Simplifying requires examining both the numerator \(13x + 14\) and the denominator \((2x+1)(3x+4)\) for common factors.
• In this case, the numerator and the denominator share no common factors.

Thus, the expression remains \( \frac{13x + 14}{(2x+1)(3x+4)} \), which means it is already in its simplest form.
Understanding how to identify and simplify expressions is valuable. It assists in ensuring that the results of mathematical operations are not only correct but also presented in their most straightforward form.
This makes further operations and applications easier to handle and understand.