Problem 61 - Elementary and Intermediate Algebra

Question

Simplify. See Example 7 . $$ \frac{3-4 t}{8 t-6} $$

Step-by-Step Solution

Answer

The simplified form of the expression is \( \frac{-1}{2} \).

Step 1: Identify the Numerator and Denominator

The given expression is a fraction \( \frac{3-4t}{8t-6} \). The numerator is \( 3-4t \), and the denominator is \( 8t-6 \).

Step 2: Factor the Numerator

The numerator is already in a simple form, \( 3-4t \), and cannot be factored further with integer factors.

Step 3: Factor the Denominator

For the denominator \( 8t-6 \), factor out the greatest common factor, which is 2. This gives us \( 8t-6 = 2(4t-3) \).

Step 4: Simplify the Expression

The expression is \( \frac{3-4t}{2(4t-3)} \). Rewriting this using distributive property, we have \( \frac{-(4t-3)}{2(4t-3)} \). The \( 4t-3 \) terms cancel each other out, resulting in \( \frac{-1}{2} \).

Step 5: Write the Final Simplified Form

The fraction simplifies to \( \frac{-1}{2} \). This is the simplest form of the given rational expression.

Key Concepts

Rational Expressions

Rational expressions are fractions that have polynomials in both the numerator and the denominator. They can be considered as algebraic fractions that require careful manipulation to simplify. In our example, the expression \(\frac{3-4t}{8t-6}\) is a rational expression. It resembles a standard fraction, but instead of numbers, it contains polynomial terms.To work with rational expressions effectively, it's essential to understand:

• The basics of polynomial operations, such as addition, subtraction, multiplication, and factoring.
• How to identify the numerator and denominator in the given expression and to assess if they can be factored further.
• The importance of simplifying these expressions to make complex algebraic operations more manageable.

When simplifying a rational expression, like \(\frac{3-4t}{8t-6}\), our goal is to transform it into its most reduced form while maintaining its equivalent rational value.

Greatest Common Factor

The Greatest Common Factor (GCF) is a key concept when simplifying algebraic expressions, including rational expressions. By identifying the GCF, we can effectively reduce fractions by factoring out this commonality. When given a polynomial, the GCF is usually the largest expression that divides all the terms without a remainder.For instance, in the expression \(8t-6\):

• Both terms are divisible by 2, which is their GCF. Factoring out 2 simplifies the expression to \(2(4t-3)\).
• Finding the GCF involves recognizing common numerical and variable factors that can simplify the polynomial's structure.

By factoring out the GCF, we make the expression simpler and ready for the next stages of simplification.

Canceling Terms

In algebraic fractions, canceling terms is the process of simplifying expressions by removing common factors in the numerator and denominator. It's an important step after factorization because it reduces the fraction to its simplest form.Consider the fraction \(\frac{-(4t-3)}{2(4t-3)}\):

• The terms \(4t-3\) appear in both numerator and denominator, hence, they can be canceled out.
• This leaves us with a much simplified expression, \(\frac{-1}{2}\), because the \(4t-3\) terms negate each other when divided.
• Cancelation is valid because it's akin to dividing both the top and bottom of the fraction by the same non-zero factor, preserving the expression's equivalency.

When canceling terms, ensure you are only removing whole factors, not altering or wrongly simplifying the polynomials involved. This practice keeps your solutions mathematically sound and accurate.