Simplify (5x-5)/(6x-12)

The question is asking for the simplification of a rational expression (also known as an algebraic fraction). In the provided expression, (5x-5)/(6x-12), the task is to reduce the expression to its simplest form by factoring common factors in the numerator (top part of the fraction) and denominator (bottom part of the fraction) and cancelling them out if possible. The simplification process may include factoring out common terms or applying properties of algebraic expressions to simplify the expression to the lowest terms.

$\frac{5 x - 5}{6 x - 12}$

Answer

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Solution:

Step 1: Factor out the common factor from the numerator

Extract the common factor of 5 from the terms in the numerator.

Step 1.1

Take out the factor of 5 from $5 x$ to get $\frac{5 (x) - 5}{6 x - 12}$ .

Step 1.2

Remove the factor of 5 from $- 5$ to obtain $\frac{5 (x) + 5 (- 1)}{6 x - 12}$ .

Step 1.3

Complete the factoring in the numerator to express it as $\frac{5 (x - 1)}{6 x - 12}$ .

Step 2: Factor out the common factor from the denominator

Identify and factor out the common factor of 6 from the terms in the denominator.

Step 2.1

Factor out the 6 from $6 x$ to rewrite the expression as $\frac{5 (x - 1)}{6 (x) - 12}$ .

Step 2.2

Remove the factor of 6 from $- 12$ to get $\frac{5 (x - 1)}{6 x + 6 \cdot (- 2)}$ .

Step 2.3

Finalize the factoring in the denominator to simplify it to $\frac{5 (x - 1)}{6 (x - 2)}$ .

Knowledge Notes:

To simplify a rational expression like $\frac{5 x - 5}{6 x - 12}$ , we look for common factors in the numerator and the denominator that can be factored out. Factoring is the process of expressing an algebraic expression as a product of its factors.

Factoring out common terms: When each term in a polynomial shares a common factor, we can factor it out using the distributive property. For example, $a x + a y = a (x + y)$ .
Simplifying rational expressions: After factoring, we can simplify a rational expression by canceling out common factors in the numerator and the denominator, as long as they are not zero.
Restrictions in rational expressions: When simplifying, we must consider the values for which the original expression is undefined (e.g., values that make the denominator zero). These restrictions should be noted even after simplification.

In this problem, we factored out 5 from the numerator and 6 from the denominator. This process did not change the values for which the expression is defined, since the original denominator $6 x - 12$ is zero when $x = 2$ , and the simplified denominator $6 (x - 2)$ is also zero when $x = 2$ . Thus, the simplification is valid for all $x$ except $x = 2$ .