Problem 33
Question
Simplify each complex rational expression. $$ \frac{x-\frac{4}{3 x-1}}{1-\frac{2 x-2}{3 x-1}} $$
Step-by-Step Solution
Verified Answer
$
\frac{x-\frac{4}{3x-1}}{1-\frac{2x-2}{3x-1}}
$
Answer: The simplified form of the complex rational expression is $\frac{3x^2 - x - 4}{x + 1}$.
1Step 1: Identify the Least Common Denominator (LCD)
The LCD is the smallest multiple that both denominators (in this case, 3x - 1) share. Since there are no other denominators to compare, our LCD is simply 3x - 1.
2Step 2: Multiply the numerator and the denominator by the LCD
We will multiply both the numerator and the denominator by (3x - 1), to eliminate the fractions.
$$
\frac{x-\frac{4}{3x-1}}{1-\frac{2x-2}{3x-1}} \cdot \frac{3x - 1}{3x - 1}
$$
3Step 3: Distribute the LCD to each term
Now, we distribute (3x - 1), and we will obtain:
$$
\frac{(x(3x-1)) - 4}{(1(3x-1)) - (2x-2)}
$$
4Step 4: Simplify expressions
Further simplify the expression by expanding the terms and combining like terms.
In the numerator, we have:
$$
(3x^2 - x) - 4
$$
The new numerator is:
$$
3x^2 - x - 4
$$
In the denominator, we have:
$$
(3x - 1) - (2x - 2)
$$
To simplify, we will distribute the negative sign and then combine like terms:
$$
3x - 1 - 2x + 2
$$
The new denominator is:
$$
x + 1
$$
5Step 5: Rewriting the simplified complex rational expression
Our final simplified complex rational expression is given by:
$$
\frac{3x^2 - x - 4}{x + 1}
$$
Key Concepts
SimplificationCommon DenominatorRational ExpressionsAlgebraic Expressions
Simplification
Simplifying a complex rational expression means making it as simple as possible. Complex rational expressions contain fractions over fractions. The goal is to rewrite them so they don't have fractions within the numerator or denominator.
To simplify complex rational expressions, you often need to:
In our original problem, the numerator and denominator each had terms that were fractions. By multiplying by the LCD, we eliminated the fractions, leading to a much simpler final expression.
To simplify complex rational expressions, you often need to:
- Identify the Least Common Denominator (LCD).
- Multiply both the numerator and denominator by this LCD to eliminate fractions.
- Distribute, expand, and simplify the expression by combining like terms.
In our original problem, the numerator and denominator each had terms that were fractions. By multiplying by the LCD, we eliminated the fractions, leading to a much simpler final expression.
Common Denominator
Finding a common denominator allows us to manipulate fractions so they can be added, subtracted, or simplified easily. In algebraic expressions, it's typically referred to as the Least Common Denominator (LCD). It is crucial for removing fractions from complex rational expressions.
To find the LCD, identify the smallest expression that each denominator can divide into without leaving a remainder. In the given exercise, the common denominator is \(3x - 1\). It is used for both the numerator and denominator of the entire complex fraction so that the fractions are eliminated during simplification.
The process ensures that each fraction is multiplied correctly, which assists in combining or canceling terms smoothly.
To find the LCD, identify the smallest expression that each denominator can divide into without leaving a remainder. In the given exercise, the common denominator is \(3x - 1\). It is used for both the numerator and denominator of the entire complex fraction so that the fractions are eliminated during simplification.
The process ensures that each fraction is multiplied correctly, which assists in combining or canceling terms smoothly.
Rational Expressions
Rational expressions are fractions in which both the numerator and the denominator are polynomials. These expressions behave similarly to regular fractions, but they involve variables.
When working with rational expressions, understanding polynomial operations, such as factoring, expanding, and reducing terms, is essential. Managing these expressions often involves simplifying them to their most reduced form.
When working with rational expressions, understanding polynomial operations, such as factoring, expanding, and reducing terms, is essential. Managing these expressions often involves simplifying them to their most reduced form.
- You can add, subtract, multiply, and divide rational expressions using similar techniques applied to simple fractions.
- Being mindful of restrictions on the variable is necessary, as dividing by zero is undefined in mathematics.
Algebraic Expressions
Algebraic expressions consist of numbers, variables, and operation signs. They can represent real-world problems or abstract concepts in mathematics.
Key operations on algebraic expressions include addition, subtraction, multiplication, division, and exponentiation. Becoming proficient in manipulating these expressions is crucial for solving complex problems.
Key operations on algebraic expressions include addition, subtraction, multiplication, division, and exponentiation. Becoming proficient in manipulating these expressions is crucial for solving complex problems.
- Combining like terms, factoring, and expanding are common techniques.
- Identifying patterns, such as perfect squares or difference of squares, helps in simplifying expressions efficiently.
Other exercises in this chapter
Problem 33
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