Problem 37 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \frac{x+8}{3 x-1}+\frac{x+3}{x+1} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{4x^2+11x+5}{(3x-1)(x+1)}\).

Step 1: Identify the LCD

Identify the least common denominator (LCD) of the given fractions. Because the denominators \(3x-1\) and \(x+1\) have no factors in common, their LCD is simply their product, \(3x-1\) times \(x+1\).

Step 2: Rewrite fractions

Rewrite fractions using the LCD. To do this, multiply the numerator and denominator of each fraction by the missing factor from the LCD. \n\n\(\frac{x+8}{3x-1} * \frac{x+1}{x+1} + \frac{x+3}{x+1} * \frac{3x-1}{3x-1}\)

Step 3: Simplify each fraction

Now you should apply the FOIL method (First, Inside, Outer, Last) to each pair of binomials to simplify the terms.\n\n\(\frac{(x+1)(x+8)}{(3x-1)(x+1)} + \frac{(3x-1)(x+3)}{(x+1)(3x-1)}\)

Step 4: Combine the fractions

The denominators are now the same, so you can combine the fractions.\n\n\(\frac{(x+1)(x+8)+(3x-1)(x+3)}{(3x-1)(x+1)}\)

Step 5: Simplify the numerator

Now, you can expand every term in the numerator.\n\n\(\frac{x^2+8x+x+8+3x^2-1x+3x-3}{(3x-1)(x+1)}\)

Step 6: Combine Like Terms

Combine Like Terms in the numerator to get the final simplified expression. This involves grouping and adding together 'like' terms - those which have the same variable to the same power.\n\n\(\frac{4x^2+11x+5}{(3x-1)(x+1)}\)

Key Concepts

Simplification

Simplification in algebra involves reducing expressions or equations to their simplest form, making them easier to understand and solve. Being able to simplify expressions is a critical skill in mathematics that helps with solving equations and problem-solving.
When simplifying, the goal is to condense the expression so that it has the fewest possible terms and operations.

• Always look for common factors to factor expressions.
• Be sure to combine like terms, which means adding or subtracting terms that have identical variable parts.
• Remember that the simplest form of an expression is not always a single number; it might be a more compact algebraic expression.

Simplifying expressions makes calculations smoother and solutions clearer. It also allows easy recognition of equivalent expressions, which are different forms of the same expression.

LCD (Least Common Denominator)

The Least Common Denominator (LCD) is crucial when dealing with fractions that have different denominators. It is the smallest expression that is a multiple of each of the denominators in a set of fractions.
To find the LCD, it's important to:

• Identify the denominators of each fraction.
• Factor each denominator if necessary.
• Determine the LCD by taking the product of each unique factor, raised to the highest power that appears in the denominators.

In the simplification process, the LCD serves as a tool to rewrite fractions with unlike denominators as equivalent fractions with the same denominator. This allows for straightforward addition or subtraction between the fractions, as seen in the exercise where fractions \( \frac{x+8}{3x-1} \) and \( \frac{x+3}{x+1} \) are rewritten utilizing their combined denominator \((3x-1)(x+1)\).

Binomials

Binomials are algebraic expressions that consist of exactly two terms. They form an essential part of polynomial algebra and appear frequently in mathematics. Examples include expressions like \(x + 8\) or \(3x - 1\). Handling binomials involves being comfortable with their structure and operations involving them.

When working with binomials, you will often perform operations such as addition, subtraction, and multiplication. Understanding how binomials interact is key to simplifying complex expressions.

• Addition/Subtraction: Combine by adding or subtracting the corresponding coefficients.
• Multiplication: Use the FOIL method, which is described in the next section.
• Factoring: Look for common factors or special products (e.g., difference of squares).

Mastery of binomials greatly aids your ability to simplify and solve algebraic expressions efficiently.

FOIL Method

The FOIL Method is a technique used to expand the product of two binomials. It's an acronym that stands for First, Outside, Inside, Last, representing the order in which you multiply the terms of each binomial.
This method simplifies the task of multiplying two binomials, ensuring that all components are included and none are overlooked. Here's a breakdown of how to apply FOIL:

• First: Multiply the first terms of each binomial.
• Outside: Multiply the outer terms in the binomials.
• Inside: Multiply the inner terms of the binomials.
• Last: Multiply the last terms of each binomial.

Once multiplied, these components are combined to form a simplified expression that represents the product of the binomials. For instance, when expanding \((x+1)(x+8)\) and \((3x-1)(x+3)\), using FOIL is efficient and systematically thorough. By completing these steps, you ensure the expression simplifies correctly, leading to the final form provided in the given exercise.