To simplify the given expression [tex]\(\frac{6x + 6}{3x}\)[/tex], we can follow these steps:
1. Factor the numerator:
The numerator [tex]\(\(6x + 6\)[/tex]\) can be factored by taking out the common factor which is 6.
[tex]\[
6x + 6 = 6(x + 1)
\][/tex]
2. Rewrite the fraction using the factored form of the numerator:
Substituting the factored form into the original expression, we get:
[tex]\[
\frac{6(x + 1)}{3x}
\][/tex]
3. Separate the constants:
We can simplify the fraction by dividing out the constants. We see that 6 in the numerator and 3 in the denominator have a common factor, which is 3.
[tex]\[
\frac{6(x + 1)}{3x} = \frac{6}{3} \cdot \frac{(x + 1)}{x}
\][/tex]
Simplify [tex]\(\frac{6}{3}\)[/tex]:
[tex]\[
\frac{6}{3} = 2
\][/tex]
4. Multiply the simplified constants by the remaining fraction:
Now the expression is:
[tex]\[
2 \cdot \frac{x + 1}{x}
\][/tex]
5. Separate the terms in the fraction:
We can split the fraction [tex]\(\frac{x + 1}{x}\)[/tex] into two separate fractions:
[tex]\[
\frac{x + 1}{x} = \frac{x}{x} + \frac{1}{x}
\][/tex]
Simplify [tex]\(\frac{x}{x}\)[/tex] to 1:
[tex]\[
\frac{x}{x} = 1
\][/tex]
Thus,
[tex]\[
\frac{x + 1}{x} = 1 + \frac{1}{x}
\][/tex]
6. Combine the simplified parts:
Now multiply 2 by the simplified fraction:
[tex]\[
2 \cdot \left(1 + \frac{1}{x}\right)
\][/tex]
7. Distribute the multiplication:
Distribute the 2 across the sum within the parentheses:
[tex]\[
2 \cdot 1 + 2 \cdot \frac{1}{x} = 2 + \frac{2}{x}
\][/tex]
Thus, the simplified form of the given expression [tex]\(\frac{6x + 6}{3x}\)[/tex] is:
[tex]\[
2 + \frac{2}{x}
\][/tex]
