Problem 28 - Algebra for College Students

Question

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{x+1}{4}+\frac{x-3}{6}-\frac{x-2}{8} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{7x}{24} \).

Step 1: Identify the LCM of the Denominators

The denominators of the expressions are 4, 6, and 8. First, find the least common multiple (LCM) of these numbers. The LCM of 4, 6, and 8 is 24.

Step 2: Rewrite Each Expression with the LCM as Denominator

Convert each term to have the common denominator of 24. For \( \frac{x+1}{4} \), multiply both the numerator and denominator by 6 to get \( \frac{6(x+1)}{24} \). For \( \frac{x-3}{6} \), multiply both the numerator and denominator by 4 to get \( \frac{4(x-3)}{24} \). For \( \frac{x-2}{8} \), multiply both the numerator and denominator by 3 to get \( \frac{3(x-2)}{24} \).

Step 3: Combine the Expressions into a Single Fraction

Since all the expressions now have the same denominator, combine them: \[ \frac{6(x+1)}{24} + \frac{4(x-3)}{24} - \frac{3(x-2)}{24} = \frac{6(x+1) + 4(x-3) - 3(x-2)}{24} \].

Step 4: Distribute and Simplify the Numerator

Distribute the numerical coefficients in each term in the numerator: \[ 6(x+1) = 6x + 6 \], \[ 4(x-3) = 4x - 12 \], and \[ 3(x-2) = 3x - 6 \]. Substitute these back into the expression to get \[ \frac{6x + 6 + 4x - 12 - 3x + 6}{24} \].

Step 5: Combine Like Terms in the Numerator

Combine the like terms in the numerator: \( (6x + 4x - 3x) = 7x \) and \( (6 - 12 + 6) = 0 \). The expression simplifies to \( \frac{7x}{24} \).

Step 6: Express the Rational Expression in Simplest Form

The simplified expression is already \( \frac{7x}{24} \), which is in its simplest form since 7 and 24 have no common factors other than 1.

Key Concepts

Least Common Multiple (LCM)

Understanding the concept of the Least Common Multiple (LCM) is fundamental when working with fractions, especially in rational expressions. The LCM of a set of numbers is the smallest number that is a multiple of each number in the group. In our given problem, we have denominators 4, 6, and 8.
To find the LCM, we look for the smallest number that evenly divides each of these denominators.

• Start by identifying the prime factors of each number: 4 (2 × 2), 6 (2 × 3), and 8 (2 × 2 × 2).
• Take the highest power of each prime number present: 2 (cubed) from 8 and 3 from 6.
• Multiply these together: 2³ × 3 = 24.

Thus, the LCM of 4, 6, and 8 is 24.
The LCM provides us with a common base that can be applied to each fraction, allowing us to add or subtract them easily.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. Let's illustrate this with our example.

• Once each fraction in the expression: \( \frac{x+1}{4}, \frac{x-3}{6}, \text{and} \frac{x-2}{8} \) is rewritten to have the same denominator (24), the next step is to add or subtract the numerators.

Combining the terms becomes straightforward, as they all share the LCM as the denominator:

• \( \frac{6(x+1)}{24} + \frac{4(x-3)}{24} - \frac{3(x-2)}{24} \)
• After combining like terms, the expression simplifies to \( \frac{7x}{24} \).

In this result, the numerator 7x and the denominator 24 share no common factors, indicating that the fraction is in its simplest form.

Polynomial Operations

To manage rational expressions efficiently, mastering polynomial operations like distribution and combining like terms is essential. In our given expression, we performed these operations during the simplification process.
Distributing numbers through polynomials changes forms without altering values:

• For \( 6(x+1) \), multiply both terms inside the parentheses by 6, resulting in \( 6x + 6 \).
• Applying the same operation to \( 4(x-3) \), results in \( 4x - 12 \).

Once distributed, we combined like terms.

• Select all x terms together: \( 6x + 4x - 3x = 7x \).
• Gather constant terms: \( 6 - 12 + 6 = 0 \).

Finally, our operation results in the simplified rational expression \( \frac{7x}{24} \). Understanding these polynomial operations aids in expressing other similar rational expressions in their simplest forms.