Simplify (3x-6)/(9x+18)

The question asks you to perform algebraic simplification on the given rational expression. This process involves reducing the expression to its simplest form by factoring and canceling out any common terms in the numerator and the denominator.

$\frac{3 x - 6}{9 x + 18}$

Answer

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Solution:

Simplify the expression $\frac{3 x - 6}{9 x + 18}$

Step 1: Identify and factor out common factors

Step 1.1: Extract the factor of $3$ from $3 x$ in the numerator.

$\frac{3 (x) - 6}{9 x + 18}$
Step 1.2: Extract the factor of $3$ from $- 6$ in the numerator.

$\frac{3 x - 3 \cdot 2}{9 x + 18}$
Step 1.3: Recognize that both terms in the numerator share a factor of $3$ .

$\frac{3 (x - 2)}{9 x + 18}$
Step 1.4: Proceed to cancel out the common factors.
- Step 1.4.1: Factor out $3$ from $9 x$ in the denominator.
  
  $\frac{3 (x - 2)}{3 (3 x) + 18}$
- Step 1.4.2: Factor out $3$ from $18$ in the denominator.
  
  $\frac{3 (x - 2)}{3 (3 x) + 3 \cdot 6}$
- Step 1.4.3: Notice that the entire denominator has a common factor of $3$ .
  
  $\frac{3 (x - 2)}{3 (3 x + 6)}$
- Step 1.4.4: Eliminate the common factor of $3$ from both numerator and denominator.
  
  $\frac{3 (x - 2)}{3 (3 x + 6)}$
- Step 1.4.5: Simplify the expression after cancellation.
  
  $\frac{x - 2}{3 x + 6}$

Step 2: Factor out common factors in the simplified denominator

Step 2.1: Extract the factor of $3$ from $3 x$ in the denominator.

$\frac{x - 2}{3 (x) + 6}$
Step 2.2: Extract the factor of $3$ from $6$ in the denominator.

$\frac{x - 2}{3 x + 3 \cdot 2}$
Step 2.3: Recognize the common factor of $3$ in the entire denominator.

$\frac{x - 2}{3 (x + 2)}$

Knowledge Notes:

To simplify a rational expression, you should follow these steps:

Factorization: Break down both the numerator and the denominator into their prime factors or common factors.
Cancellation: Cancel out common factors that appear in both the numerator and the denominator.
Simplify: After canceling, rewrite the expression in its simplest form.

In this problem, we identified that both the numerator and the denominator have a common factor of $3$ . By factoring out this common factor, we were able to cancel it from both parts of the fraction, simplifying the expression.

It's important to remember that you can only cancel factors that are multiplied together, not terms that are added or subtracted. Always ensure that the terms you are canceling are indeed factors before proceeding with the cancellation.