Answer :
To solve the problem, let's break it down step-by-step.
Understanding Inverses:
- The additive inverse of a number is what you add to the number to get zero. For example, the additive inverse of [tex]\frac{5}{6}[/tex] is [tex]-\frac{5}{6}[/tex].
- The multiplicative inverse of a number is what you multiply the number by to get one. For example, the multiplicative inverse of [tex]a[/tex] (where [tex]a \neq 0[/tex]) is [tex]\frac{1}{a}[/tex].
Find the Multiplicative Inverse:
- First, evaluate [tex]\frac{-5}{7} \times \frac{14}{15}[/tex].
- Multiplying the fractions:
[tex]\frac{-5}{7} \times \frac{14}{15} = \frac{-5 \times 14}{7 \times 15} = \frac{-70}{105}.[/tex] - Simplify [tex]\frac{-70}{105}[/tex]. The greatest common divisor of 70 and 105 is 35, so:
[tex]\frac{-70}{105} = \frac{-70 \div 35}{105 \div 35} = \frac{-2}{3}.[/tex] - The multiplicative inverse of [tex]\frac{-2}{3}[/tex] is [tex]\frac{-3}{2}[/tex].
Subtract the Additive Inverse:
- Now, subtract the additive inverse of [tex]\frac{5}{6}[/tex], which is [tex]-\frac{5}{6}[/tex], from the multiplicative inverse [tex]\frac{-3}{2}[/tex].
- This computation is as follows:
[tex]\frac{-3}{2} - (-\frac{5}{6}) = \frac{-3}{2} + \frac{5}{6}.[/tex] - To add these fractions, find a common denominator. The least common denominator of 2 and 6 is 6.
- Convert [tex]\frac{-3}{2}[/tex] to [tex]\frac{-9}{6}[/tex] (because [tex]-3 \times 3 = -9[/tex] and [tex]2 \times 3 = 6[/tex]).
- Add [tex]\frac{-9}{6} + \frac{5}{6}[/tex]:
[tex]\frac{-9}{6} + \frac{5}{6} = \frac{-9 + 5}{6} = \frac{-4}{6}.[/tex] - Simplify [tex]\frac{-4}{6}[/tex] by dividing both numerator and denominator by 2:
[tex]\frac{-4 \div 2}{6 \div 2} = \frac{-2}{3}.[/tex]
The result, [tex]\frac{-2}{3}[/tex], is the final answer.
