Problem 59
Question
simplify each complex rational expression. $$ \frac{\frac{x}{3}-1}{x-3} $$
Step-by-Step Solution
Verified Answer
The simplified expression is \( \frac{1}{3} \).
1Step 1: Remove Fraction from Denominator
Multiply the numerator and denominator by 3 to get rid of the fraction in the numerator. This results in the expression: \[ \frac{3(\frac{x}{3} - 1)}{3(x - 3)} \]
2Step 2: Simplify Expression
Simplify the expression further to \[ \frac{x - 3}{3x - 9} \]
3Step 3: Further Decomposition
Here it is seen that the numerator and the denominator are multiples of each other, which can be written as \[ \frac{x - 3}{3(x - 3)} \]
Key Concepts
Algebraic TechniquesRational Expressions SimplificationFractional Expressions
Algebraic Techniques
Mastering algebraic techniques is essential when working with complex rational expressions. These include various operations such as simplifying, factoring, expanding, and manipulating expressions to solve equations or simplify expressions.
For instance, in the given exercise, the technique used to simplify the complex rational expression \( \frac{\frac{x}{3}-1}{x-3} \) involves first eliminating the complex fraction by multiplying both the numerator and the denominator by 3, a common denominator for the fractions involved. This is a common technique used to remove fractions from fractions, creating a simpler expression to work with.
Similarly, recognizing patterns, such as when the numerator and denominator share common factors, allows for further simplification through factoring and cancellation. Such insights are critical in streamlining solutions and arriving at the simplest form of an expression.
For instance, in the given exercise, the technique used to simplify the complex rational expression \( \frac{\frac{x}{3}-1}{x-3} \) involves first eliminating the complex fraction by multiplying both the numerator and the denominator by 3, a common denominator for the fractions involved. This is a common technique used to remove fractions from fractions, creating a simpler expression to work with.
Similarly, recognizing patterns, such as when the numerator and denominator share common factors, allows for further simplification through factoring and cancellation. Such insights are critical in streamlining solutions and arriving at the simplest form of an expression.
Rational Expressions Simplification
Rational expressions simplification involves reducing the complexity of fractions that have polynomials in the numerator, denominator, or both. The goal is to make the expression as straightforward as possible.
In the provided example, once we've eliminated the fraction in the numerator by multiplying by 3, we simplify \( \frac{3(\frac{x}{3}-1)}{3(x-3)} \) to \( \frac{x-3}{3x-9} \). This is a significant step in simplification. However, to fully simplify the expression, we need to identify and cancel out any common factors present in both the numerator and the denominator. In doing so, we can reduce the expression further.
The final step in our example utilizes the greatest common factor (GCF), which is \( x - 3 \) in this case. Factoring out the GCF and canceling the common terms leads us to an even more simplified form, showcasing how integral these simplification techniques are in algebra.
In the provided example, once we've eliminated the fraction in the numerator by multiplying by 3, we simplify \( \frac{3(\frac{x}{3}-1)}{3(x-3)} \) to \( \frac{x-3}{3x-9} \). This is a significant step in simplification. However, to fully simplify the expression, we need to identify and cancel out any common factors present in both the numerator and the denominator. In doing so, we can reduce the expression further.
The final step in our example utilizes the greatest common factor (GCF), which is \( x - 3 \) in this case. Factoring out the GCF and canceling the common terms leads us to an even more simplified form, showcasing how integral these simplification techniques are in algebra.
Fractional Expressions
Fractional expressions, or fractions that contain algebraic expressions in their numerators, denominators, or both, are common in algebra. Handling them involves understanding fundamental fraction rules, such as finding a common denominator for addition and subtraction, as well as multiplying and dividing fractions.
In the context of the given problem, after the initial step of removing the complex fraction, we are left with a fractional expression with polynomials in the numerator and the denominator. It's imperative to understand that simplifying fractional expressions doesn't alter the value of the expression; it only presents it in its most reduced form.
The simplification step often involves factoring both the numerator and the denominator if possible and then reducing common factors. This was illustrated in the provided solution, resulting in the simplified expression \( \frac{1}{3} \), as \( x - 3 \) was canceled from both the numerator and the denominator after factoring. The ability to manipulate such fractional expressions is a valuable skill as it widely applies to various fields of mathematics and beyond.
In the context of the given problem, after the initial step of removing the complex fraction, we are left with a fractional expression with polynomials in the numerator and the denominator. It's imperative to understand that simplifying fractional expressions doesn't alter the value of the expression; it only presents it in its most reduced form.
The simplification step often involves factoring both the numerator and the denominator if possible and then reducing common factors. This was illustrated in the provided solution, resulting in the simplified expression \( \frac{1}{3} \), as \( x - 3 \) was canceled from both the numerator and the denominator after factoring. The ability to manipulate such fractional expressions is a valuable skill as it widely applies to various fields of mathematics and beyond.
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Problem 58
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