Problem 57 - Elementary Algebra

Question

For the following problems, reduce each rational expression if possible. If not possible, state the answer in lowest terms. \(\frac{4 x-7}{-7}\)

Step-by-Step Solution

Answer

Answer: The simplified expression in its lowest terms is \(\frac{-4x+7}{7}\).

Step 1: Check for common factors

In the given expression, \(\frac{4x-7}{-7}\), the numerator is a binomial and the denominator is a constant. There are no common factors between the two, so we cannot cancel any terms out.

Step 2: Simplify the expression in lowest terms

We can represent the rational expression in its lowest terms by simplifying the sign of the expression. Since the denominator is negative, we can change the signs of both the numerator and the denominator to get: $$\frac{-(4x-7)}{-(-7)}$$ This can be further simplified as: $$\frac{-4x+7}{7}$$ So, the rational expression in its lowest terms is \(\frac{-4x+7}{7}\).

Key Concepts

Algebra

Algebra is the branch of mathematics that deals with symbols and the rules for manipulating those symbols. In this exercise, we are working with algebraic expressions, specifically rational expressions. Rational expressions involve fractions where the numerator and the denominator are polynomials. Algebra allows us to manipulate these expressions and simplify them by applying different mathematical operations like addition, subtraction, multiplication, and division. This manipulation is guided by a set of rules that help maintain equality.
Algebra comes in handy when simplifying expressions like \( \frac{4x-7}{-7} \) because it helps us understand how operations on terms can transform the expression into its simplest form.

Simplification

Simplification is the process of reducing a mathematical expression to its simplest form. When dealing with rational expressions, simplification often involves finding common factors, reducing terms, or changing the signs to make the expression more straightforward.
In the exercise, we seek to simplify \( \frac{4x-7}{-7} \) to remove negative signs that can make the expression confusing. This is achieved by multiplying both the numerator and the denominator by \(-1\), which changes the expression to \( \frac{-4x+7}{7} \).
Simplification:

• Ensures expressions are neat and manageable.
• Makes it easier to understand and work with the expression in complex operations.

This step is crucial in algebra, especially in making further calculations more accessible and error-free.

Numerator and Denominator

In a rational expression, the numerator and the denominator are two key parts. They resemble the top and bottom parts of a fraction. The numerator in our example is a binomial, \(4x-7\), and the denominator is a constant, \(-7\).
Both parts serve specific functions:

• The numerator denotes the expression or value being divided.
• The denominator tells how many equal parts make up a whole or define the number of parts to which the numerator is compared.

Understanding these components allows us to perform operations such as simplifying, factoring, or even expanding expressions. When simplifying, we often look for common factors between these two parts that can be simplified or canceled out.

Binomials

A binomial is an algebraic expression with exactly two terms. In the given rational expression, \(4x-7\) is a binomial, consisting of the linear term \(4x\) and the constant term \(-7\). Binomials are crucial in algebra due to their frequent appearance in polynomial operations and factorizations.
When dealing with binomials, you may observe some of the following operations:

• Adding or subtracting binomials.
• Multiplying binomials, often using distribution or FOIL method.
• Dividing polynomials where binomials may serve as numerators or denominators.

Recognizing a binomial in the numerator, as in our example, provides insight into how the expression interacts with other components. This becomes especially helpful in tasks like factoring or simplifying where binomials play a critical role.