To solve the equation [tex]\(\frac{3 \pm \sqrt{3}}{3} = 0\)[/tex], let's examine each component of the equation step-by-step.
1. Split the Equation into Two Separate Equations:
The equation [tex]\(\frac{3 \pm \sqrt{3}}{3} = 0\)[/tex] can be split into two distinct equations:
[tex]\[
\frac{3 + \sqrt{3}}{3} = 0
\][/tex]
and
[tex]\[
\frac{3 - \sqrt{3}}{3} = 0
\][/tex]
2. Solve Each Equation Separately:
Let's solve each of the above equations one by one.
For the first equation:
[tex]\[
\frac{3 + \sqrt{3}}{3} = 0
\][/tex]
To clear the fraction, multiply both sides of the equation by 3:
[tex]\[
3 + \sqrt{3} = 0
\][/tex]
Now, isolate [tex]\(\sqrt{3}\)[/tex] by subtracting 3 from both sides:
[tex]\[
\sqrt{3} = -3
\][/tex]
Notice that [tex]\(\sqrt{3}\)[/tex] represents the principal square root of 3, which is always a positive number (approximately 1.732). Therefore, this equation has no solution since a positive number cannot equal a negative one.
For the second equation:
[tex]\[
\frac{3 - \sqrt{3}}{3} = 0
\][/tex]
Similarly, multiply both sides of the equation by 3 to clear the fraction:
[tex]\[
3 - \sqrt{3} = 0
\][/tex]
Isolate [tex]\(\sqrt{3}\)[/tex] by subtracting 3 from both sides:
[tex]\[
\sqrt{3} = 3
\][/tex]
Again, since [tex]\(\sqrt{3}\)[/tex] (which is approximately 1.732) is always a positive number and less than 3, there is no solution to this equation as well.
Since both equations have been demonstrated to have no solutions, we can conclude that the original equation [tex]\(\frac{3 \pm \sqrt{3}}{3} = 0\)[/tex] has no solutions.
Thus, the solution set is:
[tex]\[
\boxed{[]}
\][/tex]
