To simplify the expression [tex]\(\left(-\frac{1}{5} r - 4 - \frac{2}{3} r\right) - \left(-\frac{4}{5} r + 9\right)\)[/tex], let’s follow these steps methodically:
1. Distribute and Combine Like Terms:
- First, distribute the subtraction over the second group of terms:
[tex]\[
\left(-\frac{1}{5} r - 4 - \frac{2}{3} r\right) - \left(-\frac{4}{5} r + 9\right) = -\frac{1}{5} r - 4 - \frac{2}{3} r - (-\frac{4}{5} r) - 9
\][/tex]
2. Simplify the Distribution:
- Notice how subtracting a negative is equivalent to adding:
[tex]\[
-\frac{1}{5} r - 4 - \frac{2}{3} r + \frac{4}{5} r - 9
\][/tex]
3. Combine Like Terms:
- Combine the terms involving [tex]\(r\)[/tex]:
[tex]\[
-\frac{1}{5} r - \frac{2}{3} r + \frac{4}{5} r
\][/tex]
- Calculate the coefficients:
[tex]\[
-\frac{1}{5} + \frac{4}{5} - \frac{2}{3}
\][/tex]
- First, combine [tex]\(-\frac{1}{5}\)[/tex] and [tex]\(\frac{4}{5}\)[/tex]:
[tex]\[
-\frac{1}{5} r + \frac{4}{5} r = \frac{3}{5} r
\][/tex]
- Then add the remaining term [tex]\(-\frac{2}{3}\)[/tex]:
To combine these, find a common denominator, which is 15:
[tex]\[
\frac{3}{5} = \frac{9}{15} \quad \text{and} \quad -\frac{2}{3} = -\frac{10}{15}
\][/tex]
[tex]\[
\frac{9}{15} r - \frac{10}{15} r = -\frac{1}{15} r
\][/tex]
4. Combine the constant terms:
[tex]\[
-4 - 9 = -13
\][/tex]
5. Write the final simplified expression:
[tex]\[
-\frac{1}{15} r - 13
\][/tex]
After following these steps, we find that the simplified form of the expression is:
[tex]\[
-\frac{1}{15} r - 13
\][/tex]
In relation to the given choices, the correct answer is:
[tex]\[
-\frac{1}{15} r + (-13)
\][/tex]
So, the correct option is [tex]\( \boxed{-\frac{1}{15} r + (-13)} \)[/tex].
