Absolutely, let's work through this step-by-step.
1. Identify the Reciprocals and Additive Inverse:
- Reciprocal 1: [tex]\(\frac{-5}{2}\)[/tex]
- Reciprocal 2: [tex]\(\frac{4}{35}\)[/tex]
- Additive Inverse: [tex]\(\frac{3}{2}\)[/tex]
2. Multiply the Reciprocals:
- To find the product of [tex]\(\frac{-5}{2}\)[/tex] and [tex]\(\frac{4}{35}\)[/tex], we multiply the numerators and the denominators:
[tex]\[
\left(\frac{-5}{2}\right) \times \left(\frac{4}{35}\right) = \frac{-5 \times 4}{2 \times 35} = \frac{-20}{70}
\][/tex]
- Simplify the fraction [tex]\(\frac{-20}{70}\)[/tex]. Both the numerator and the denominator can be divided by their greatest common divisor, which is 10:
[tex]\[
\frac{-20}{70} = \frac{-20 \div 10}{70 \div 10} = \frac{-2}{7}
\][/tex]
3. Add the Additive Inverse:
- Now we need to add the additive inverse [tex]\(\frac{3}{2}\)[/tex] to the product [tex]\(\frac{-2}{7}\)[/tex]:
[tex]\[
\frac{-2}{7} + \frac{3}{2}
\][/tex]
- To add these fractions, we need a common denominator. The common denominator for 7 and 2 is 14. Convert both fractions:
[tex]\[
\frac{-2}{7} = \frac{-2 \times 2}{7 \times 2} = \frac{-4}{14}
\][/tex]
[tex]\[
\frac{3}{2} = \frac{3 \times 7}{2 \times 7} = \frac{21}{14}
\][/tex]
- Add the two fractions with the common denominator:
[tex]\[
\frac{-4}{14} + \frac{21}{14} = \frac{-4 + 21}{14} = \frac{17}{14}
\][/tex]
Hence, the result of multiplying the reciprocals [tex]\(\frac{-5}{2}\)[/tex] and [tex]\(\frac{4}{35}\)[/tex], and then adding the additive inverse [tex]\(\frac{3}{2}\)[/tex], is [tex]\(\frac{17}{14}\)[/tex] or as a decimal approximately [tex]\(1.2142857142857144\)[/tex].
This concludes the step-by-step solution for the given problem.
