To find the domain of the rational expression [tex]\(\frac{x-5}{3x-6}\)[/tex], we need to identify any values of [tex]\(x\)[/tex] that would cause the denominator to be zero, as division by zero is undefined in mathematics.
1. We start with the expression:
[tex]\[
\frac{x-5}{3x-6}
\][/tex]
2. Focus on the denominator [tex]\(3x-6\)[/tex]. We need to find the values of [tex]\(x\)[/tex] that make [tex]\(3x-6=0\)[/tex].
3. Solve the equation [tex]\(3x-6=0\)[/tex]:
[tex]\[
3x - 6 = 0
\][/tex]
To isolate [tex]\(x\)[/tex], add 6 to both sides:
[tex]\[
3x = 6
\][/tex]
Next, divide both sides by 3:
[tex]\[
x = 2
\][/tex]
4. [tex]\(x = 2\)[/tex] is the value that makes the denominator zero. Therefore, [tex]\(x = 2\)[/tex] is excluded from the domain because it causes the expression to be undefined.
5. Hence, the domain of the rational expression [tex]\(\frac{x-5}{3x-6}\)[/tex] includes all real numbers except [tex]\(x = 2\)[/tex].
Thus, the domain is all real numbers except [tex]\(2\)[/tex]. Therefore, the correct answer is:
- All real numbers except 2
