To perform the indicated operations and reduce the expression [tex]\(\frac{7x + 6}{x - 4} - 7\)[/tex] to its lowest terms, let's go step by step:
1. Rewrite the expression to have a common denominator:
The expression [tex]\(\frac{7x + 6}{x - 4} - 7\)[/tex] has two terms. The first term already has a denominator of [tex]\(x - 4\)[/tex], while the second term is a whole number (which can be written as [tex]\(\frac{7(x - 4)}{x - 4}\)[/tex]). Let's rewrite the whole expression with a common denominator:
[tex]\[
\frac{7x + 6}{x - 4} - 7 = \frac{7x + 6}{x - 4} - \frac{7(x - 4)}{x - 4}
\][/tex]
2. Combine the fractions:
Now that both terms have a common denominator, we can combine them into a single fraction:
[tex]\[
= \frac{7x + 6 - 7(x - 4)}{x - 4}
\][/tex]
3. Simplify the numerator:
Expand and simplify the numerator:
[tex]\[
= \frac{7x + 6 - 7x + 28}{x - 4}
\][/tex]
Combine like terms:
[tex]\[
= \frac{6 + 28}{x - 4} = \frac{34}{x - 4}
\][/tex]
Thus, the expression [tex]\(\frac{7x + 6}{x - 4} - 7\)[/tex], after performing the indicated operations and reducing it to lowest terms, simplifies to:
[tex]\[
\boxed{\frac{34}{x - 4}}
\][/tex]
