Sure, let's simplify the given expression step by step.
### Given Expression:
[tex]\[
\left(-\frac{1}{5} r - 4 - \frac{2}{3} r\right)-\left(-\frac{4}{5} r + 9\right)
\][/tex]
### Step-by-Step Simplification:
Step 1: Distribute the negative sign through the second group of terms:
[tex]\[
\left(-\frac{1}{5} r - 4 - \frac{2}{3} r\right) + \left(\frac{4}{5} r - 9\right)
\][/tex]
Step 2: Combine like terms involving [tex]\( r \)[/tex]:
Combine [tex]\(-\frac{1}{5} r\)[/tex], [tex]\(-\frac{2}{3} r\)[/tex], and [tex]\(\frac{4}{5} r\)[/tex]:
First, we find a common denominator for the fractions:
- [tex]\(\frac{1}{5} = \frac{3}{15}\)[/tex]
- [tex]\(\frac{2}{3} = \frac{10}{15}\)[/tex]
- [tex]\(\frac{4}{5} = \frac{12}{15}\)[/tex]
Then, convert these fractions to have a common denominator:
[tex]\[
-\frac{1}{5}r = -\frac{3}{15}r
\][/tex]
[tex]\[
-\frac{2}{3}r = -\frac{10}{15}r
\][/tex]
[tex]\[
\frac{4}{5}r = \frac{12}{15}r
\][/tex]
Now, combine the terms:
[tex]\[
-\frac{3}{15}r - \frac{10}{15}r + \frac{12}{15}r
\][/tex]
Combine the fractions:
[tex]\[
\left(-3 - 10 + 12\right) \frac{r}{15} = \frac{-1}{15}r
\][/tex]
So, the combined [tex]\( r \)[/tex] terms are:
[tex]\[
-\frac{1}{15}r
\][/tex]
Step 3: Combine the constant terms:
Combine [tex]\(-4\)[/tex] and [tex]\(-9\)[/tex]:
[tex]\[
-4 - 9 = -13
\][/tex]
### The Simplified Expression:
Putting it all together, the simplified expression is:
[tex]\[
-\frac{1}{15}r - 13
\][/tex]
Therefore, the correct answer is:
[tex]\[
-\frac{1}{15} r + (-13)
\][/tex]
Hence,
[tex]\[
\boxed{-\frac{1}{15} r + (-13)}
\][/tex]
This matches with one of the given options.
