Problem 7
Question
Simplify these expressions using the LCD-multiply-divide method. Compare the answers to the answers produced in Practice Set A. $$ \frac{3-\frac{1}{x}}{3+\frac{1}{x}} $$
Step-by-Step Solution
Verified Answer
Question: Simplify the expression using the LCD-multiply-divide method: $$
\frac{3x - \frac{x}{x}}{3x + \frac{x}{x}}
$$
Answer: The simplified expression is $\frac{3x - 1}{3x + 1}$.
1Step 1: Find the LCM of the denominators
The denominators we have are x and 1. The least common multiple of x and 1 is simply x.
2Step 2: Rewrite the expression in terms of LCM
We'll rewrite the given expression in terms of the LCM (x). This involves multiplying the numerators and denominators of the fractions inside the brackets by x:
$$
\frac{3x - \frac{x}{x}}{3x + \frac{x}{x}}
$$
3Step 3: Simplify the numerator and denominator
Now, simplify the numerator and denominator by removing the fractions:
$$
\frac{3x - 1}{3x + 1}
$$
4Step 4: Simplification complete
As there are no more common factors in the numerator and denominator, the simplification is complete. The simplified expression is:
$$
\frac{3x - 1}{3x + 1}
$$
Key Concepts
Least common denominator (LCD)Algebraic fractionsExpression simplificationLeast common multiple (LCM)
Least common denominator (LCD)
The least common denominator (LCD) is an instrumental concept in algebra, especially when dealing with algebraic fractions. The LCD refers to the least common multiple of the denominators of two or more fractions. It allows us to combine these fractions by converting them with denominators that are the same, thus simplifying the addition, subtraction, or comparison of fractions.
When working with algebraic fractions, the LCD is crucial because it provides a common baseline from which we can easily simplify expressions. For example, if we have two algebraic fractions with denominators of x and y, where x and y are variables or expressions, the LCD will be a multiple that includes all the variable terms present. In many cases, the LCD is just the product of the distinct denominators, but it can get more complex with more complicated expressions.
In the given exercise, the LCD helps us combine terms so that the complex fraction becomes simpler to work with and eventually reduces to a simpler form.
When working with algebraic fractions, the LCD is crucial because it provides a common baseline from which we can easily simplify expressions. For example, if we have two algebraic fractions with denominators of x and y, where x and y are variables or expressions, the LCD will be a multiple that includes all the variable terms present. In many cases, the LCD is just the product of the distinct denominators, but it can get more complex with more complicated expressions.
In the given exercise, the LCD helps us combine terms so that the complex fraction becomes simpler to work with and eventually reduces to a simpler form.
Algebraic fractions
Algebraic fractions are simply fractions that contain variables in their numerators, denominators, or both. They operate under the same principles as numerical fractions but require careful handling because of the added complexity of variables.
When simplifying algebraic fractions, we look for common factors between the numerator and denominator and cancel them out. We also often need to find the LCD when adding, subtracting or simplifying complex fractions composed of multiple algebraic fractions.
The exercise provided involves simplifying a complex algebraic fraction by first identifying and then using the LCD. By understanding how to manipulate algebraic fractions, students can tackle a wide range of problems involving polynomial expressions and equations.
When simplifying algebraic fractions, we look for common factors between the numerator and denominator and cancel them out. We also often need to find the LCD when adding, subtracting or simplifying complex fractions composed of multiple algebraic fractions.
The exercise provided involves simplifying a complex algebraic fraction by first identifying and then using the LCD. By understanding how to manipulate algebraic fractions, students can tackle a wide range of problems involving polynomial expressions and equations.
Expression simplification
Simplifying algebraic expressions is key to making complex mathematical problems more manageable. It involves reducing an expression to its most basic form while keeping its value the same. This can involve combining like terms, factoring, expanding expressions, and canceling out common factors.
To effectively simplify expressions, it's essential to have a strong grasp of fundamental algebraic principles, such as distributive property, associative property, and commutative property. In this process, we also apply operations like addition, subtraction, and multiplication to combine and reduce terms.
As demonstrated in the provided solution, the complex fraction was simplified by using the LCD, breaking down the expression, and eliminating unneeded parts, which led us to the most reduced form of the original expression.
To effectively simplify expressions, it's essential to have a strong grasp of fundamental algebraic principles, such as distributive property, associative property, and commutative property. In this process, we also apply operations like addition, subtraction, and multiplication to combine and reduce terms.
As demonstrated in the provided solution, the complex fraction was simplified by using the LCD, breaking down the expression, and eliminating unneeded parts, which led us to the most reduced form of the original expression.
Least common multiple (LCM)
The least common multiple (LCM) is closely related to the concept of the LCD. While the LCD is used to find a common base for fractions, the LCM refers to the smallest multiple that two or more numbers have in common. It is a foundational tool used not only for simplifying algebraic fractions but also for solving various problems involving multiples and divisibility.
Calculating the LCM of different numbers or expressions allows us to align them on a common scale. For instance, when resolving conflicts in scheduling or solving problems with repeating patterns, the LCM provides the frequency at which events coincide.
In the context of the exercise, the LCM serves as the LCD since we are dealing with multiples in the form of variables. The LCM determined the restructuring of the algebraic fraction before simplification could be achieved.
Calculating the LCM of different numbers or expressions allows us to align them on a common scale. For instance, when resolving conflicts in scheduling or solving problems with repeating patterns, the LCM provides the frequency at which events coincide.
In the context of the exercise, the LCM serves as the LCD since we are dealing with multiples in the form of variables. The LCM determined the restructuring of the algebraic fraction before simplification could be achieved.
Other exercises in this chapter
Problem 7
For the following problems, find the domain of each rational expression. $$ \frac{-11 x}{x^{2}-9 x+18} $$
View solution Problem 7
When the same number is added to both the numerator and denominator of the fraction \(\frac{3}{7},\) the result is \(\frac{2}{3} .\) What is the number?
View solution Problem 7
For the following problems, solve the rational equations. $$ \frac{54}{y}=\frac{27}{4} $$
View solution Problem 7
Perform each division. $$ \frac{x^{2}-4}{x^{2}+x-6} \div \frac{x^{2}+x-2}{x^{2}+4 x+3} $$
View solution