Problem 23 - Elementary Algebra

Question

For exercises 1-66, simplify. $$ \frac{3 x-6}{4 x-8} $$

Step-by-Step Solution

Answer

\( \frac{3}{4} \)

Step 1 - Factor the Numerator

Factor out the Greatest Common Factor (GCF) from the numerator. The expression for the numerator is \( 3x - 6 \). The GCF of 3 and 6 is 3.Factor out 3 from the numerator:\[ 3x - 6 = 3(x - 2) \]

Step 2 - Factor the Denominator

Factor out the GCF from the denominator. The expression for the denominator is \( 4x - 8 \). The GCF of 4 and 8 is 4.Factor out 4 from the denominator:\[ 4x - 8 = 4(x - 2) \]

Step 3 - Simplify the Fraction

Substitute the factored forms of the numerator and denominator back into the fraction:\[ \frac{3(x - 2)}{4(x - 2)} \]Since \( x - 2 \) is common to both the numerator and the denominator, cancel it out.\[ \frac{3}{4} \]

Key Concepts

Greatest Common Factor (GCF)

To simplify an algebraic fraction, the first step is to find the Greatest Common Factor (GCF) of the terms in the numerator and the denominator. The GCF is the largest number that divides evenly into each term. For example, consider the expression \( 3x - 6 \). The coefficients are 3 and 6. The GCF of 3 and 6 is 3. Similarly, for the expression \( 4x - 8 \), the GCF of 4 and 8 is 4. Identifying the GCF correctly helps in factoring expressions, making the next steps easier. Remember, always check the coefficients and constants in the terms to find their GCF.

Factoring

Factoring is the process of writing an algebraic expression as a product of its factors. Once you’ve identified the GCF, you can factor it out of each term in the expression. For instance, in the numerator \( 3x - 6 \), since the GCF is 3, we can express it as \( 3(x - 2) \). Similarly, in the denominator \( 4x - 8 \), we can factor it as \( 4(x - 2) \). Factoring makes it easier to simplify the fraction, as common factors in the numerator and denominator can be canceled.

Simplifying Fractions

After factoring the numerator and the denominator, the next step is to simplify the fraction. Substitute the factored forms back into the fraction. For example:

• Numerator: \( 3(x - 2) \)
• Denominator: \( 4(x - 2) \)

So, the fraction becomes \( \frac{3(x - 2)}{4(x - 2)} \). Since \( (x - 2) \) is common in both, it can be canceled out, resulting in \( \frac{3}{4} \). This is your simplified fraction. Simplifying fractions involves canceling out common factors, reducing the fraction to its simplest form. Always ensure the expressions are factored completely before canceling to avoid any mistakes.