Let's simplify the given expression step by step:
Given the expression:
[tex]\[
\left(-\frac{1}{8} r - 5 - \frac{1}{6} r\right) - \left(-\frac{6}{8} r + 3\right)
\][/tex]
1. Distribute the negative sign to the second group of terms within the parentheses:
[tex]\[
-\frac{1}{8} r - 5 - \frac{1}{6} r - \left(-\frac{6}{8} r + 3\right)
= -\frac{1}{8} r - 5 - \frac{1}{6} r + \frac{6}{8} r - 3
\][/tex]
2. Combine like terms with [tex]\( r \)[/tex]:
[tex]\[
-\frac{1}{8} r - \frac{1}{6} r + \frac{6}{8} r
\][/tex]
To combine these fractions, find a common denominator. The denominators are 8, 6, and 8. The least common multiple of these is 24.
Convert each fraction to have the denominator of 24:
[tex]\[
-\frac{1}{8} r = -\frac{3}{24} r, \quad -\frac{1}{6} r = -\frac{4}{24} r, \quad \text{and} \quad \frac{6}{8} r = \frac{18}{24} r
\][/tex]
Now, combine these fractions:
[tex]\[
-\frac{3}{24} r - \frac{4}{24} r + \frac{18}{24} r = \left( -3 - 4 + 18 \right) \frac{1}{24} r = \frac{11}{24} r
\][/tex]
3. Combine the constant terms:
[tex]\[
-5 - 3 = -8
\][/tex]
Putting it all together, we have:
[tex]\[
\frac{11}{24} r - 8
\][/tex]
Now, let's compare this simplified expression with the options provided:
1. [tex]\(-\frac{8}{14} r - 2\)[/tex]
2. [tex]\(\frac{11}{24} r + (-2)\)[/tex]
3. [tex]\(-\frac{4}{14} r - 8\)[/tex]
4. [tex]\(\frac{11}{24} r + (-8)\)[/tex]
The correct match is:
[tex]\[
\frac{11}{24} r + (-8)
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{4}
\][/tex]
