Sure, let's find the solution to [tex]\( -12 \frac{2}{3} - \left( -8 \frac{5}{6} \right) \)[/tex].
### Step-by-Step Solution:
1. Convert the mixed numbers to improper fractions:
- For [tex]\( -12 \frac{2}{3} \)[/tex]:
[tex]\[
-12 \frac{2}{3} = -\left(12 + \frac{2}{3}\right) = -\left(\frac{36}{3} + \frac{2}{3}\right) = -\frac{38}{3}
\][/tex]
- For [tex]\( -8 \frac{5}{6} \)[/tex]:
[tex]\[
-8 \frac{5}{6} = -\left(8 + \frac{5}{6}\right) = -\left(\frac{48}{6} + \frac{5}{6}\right) = -\frac{53}{6}
\][/tex]
2. Use the additive inverse rule:
[tex]\[
-12 \frac{2}{3} - \left( -8 \frac{5}{6} \right) = -\frac{38}{3} + \frac{53}{6}
\][/tex]
3. Find a common denominator to add these fractions:
- The common denominator of 3 and 6 is 6. Convert [tex]\(-\frac{38}{3}\)[/tex] to have a denominator of 6:
[tex]\[
-\frac{38}{3} = -\frac{38 \times 2}{3 \times 2} = -\frac{76}{6}
\][/tex]
4. Add the fractions:
[tex]\[
-\frac{76}{6} + \frac{53}{6} = \frac{-76 + 53}{6} = \frac{-23}{6}
\][/tex]
5. Convert the improper fraction back to a mixed number:
- Divide the numerator by the denominator:
[tex]\[
\frac{-23}{6} = -4 \frac{1}{6}
\][/tex]
- Here, [tex]\(-23\)[/tex] divided by 6 is [tex]\(-4\)[/tex] with a remainder of [tex]\(-5\)[/tex]. Since we usually write the remainder as a positive fraction part, [tex]\(-5\)[/tex] becomes [tex]\( 1 \)[/tex] in the context of remaining amount to infer the mixed number part. Then:
[tex]\[
\frac{-23}{6} = -4 \frac{1}{6}
\][/tex]
So the final answer is [tex]\(-4 \frac{1}{6}\)[/tex].
