To determine for which values of [tex]\( x \)[/tex] the two expressions
[tex]\[ \frac{(x-1)(x-6)}{4(x-1)} = \frac{x-6}{4} \][/tex]
are equal, let's follow a step-by-step approach to simplify and solve the equation.
1. Simplify the equation:
Let's first simplify [tex]\(\frac{(x-1)(x-6)}{4(x-1)}\)[/tex].
Notice that in the fraction [tex]\(\frac{(x-1)(x-6)}{4(x-1)}\)[/tex], the term [tex]\(x-1\)[/tex] appears in both the numerator and the denominator. If [tex]\( x \neq 1 \)[/tex], we can cancel [tex]\( x-1 \)[/tex] from both the numerator and the denominator:
[tex]\[
\frac{(x-1)(x-6)}{4(x-1)} = \frac{x-6}{4}
\][/tex]
2. Establish the condition:
After canceling [tex]\( x-1 \)[/tex], the equation simplifies to:
[tex]\[
\frac{x-6}{4} = \frac{x-6}{4}
\][/tex]
This simplified equation is true for all values of [tex]\( x \)[/tex] that do not cause a division by zero or make the cancellation invalid.
3. Identify invalid values:
It is important to remember that the cancellation of [tex]\( x-1 \)[/tex] is valid only if [tex]\( x \neq 1 \)[/tex]. If [tex]\( x = 1 \)[/tex], the original denominator would be zero:
[tex]\[
4(x-1) = 4(1-1) = 4 \cdot 0 = 0
\][/tex]
Division by zero is undefined, so [tex]\( x = 1 \)[/tex] is not a permissible solution.
4. Conclusion:
Therefore, the solution to the equation [tex]\(\frac{(x-1)(x-6)}{4(x-1)} = \frac{x-6}{4}\)[/tex] is all real numbers [tex]\( x \)[/tex] except [tex]\( x = 1 \)[/tex].
In summary:
[tex]\[ x \text{ can be any real number except } x = 1. \][/tex]
