Let's work through adding the given fractions step by step. We'll add the fractions [tex]\(-\frac{12}{7}\)[/tex] and [tex]\(\frac{3}{14}\)[/tex].
1. Find a common denominator:
The lowest common denominator (LCD) of 7 and 14 is 14.
2. Convert fractions to have the common denominator:
[tex]\(-\frac{12}{7}\)[/tex] can be converted to have a denominator of 14. We do this by multiplying both the numerator and denominator by 2:
[tex]\[
-\frac{12}{7} = -\frac{12 \times 2}{7 \times 2} = -\frac{24}{14}
\][/tex]
3. Rewrite the fractions with the common denominator:
Now we have:
[tex]\[
-\frac{24}{14} \quad \text{and} \quad \frac{3}{14}
\][/tex]
4. Add the fractions:
Since the denominators are the same, we can simply add the numerators:
[tex]\[
-\frac{24}{14} + \frac{3}{14} = \frac{-24 + 3}{14} = \frac{-21}{14}
\][/tex]
5. Simplify the result:
The fraction [tex]\(\frac{-21}{14}\)[/tex] can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 7:
[tex]\[
\frac{-21 \div 7}{14 \div 7} = \frac{-3}{2}
\][/tex]
So, the sum of [tex]\(-\frac{12}{7}\)[/tex] and [tex]\(\frac{3}{14}\)[/tex] is [tex]\(\frac{-3}{2}\)[/tex], a reduced improper fraction.
[tex]\[
\boxed{-\frac{3}{2}}
\][/tex]
