Sure, let's break this down step-by-step to find the number by which we should multiply [tex]\(\left(\frac{1}{2}\right)^{-1}\)[/tex] to get [tex]\(\left(\frac{-5}{4}\right)^{-1}\)[/tex].
1. Calculate [tex]\(\left(\frac{1}{2}\right)^{-1}\)[/tex]
Raising a fraction to the power of [tex]\(-1\)[/tex] is equivalent to taking its reciprocal.
[tex]\[
\left(\frac{1}{2}\right)^{-1} = \frac{1}{\frac{1}{2}} = 2
\][/tex]
2. Calculate [tex]\(\left(\frac{-5}{4}\right)^{-1}\)[/tex]
Again, raising a fraction to the power of [tex]\(-1\)[/tex] is equivalent to taking its reciprocal.
[tex]\[
\left(\frac{-5}{4}\right)^{-1} = \frac{1}{\frac{-5}{4}} = \frac{4}{-5} = -0.8
\][/tex]
3. Find the number that when multiplied by [tex]\(2\)[/tex] gives [tex]\(-0.8\)[/tex]
Let [tex]\(x\)[/tex] be the number we are looking for. We need to solve for [tex]\(x\)[/tex] in the equation:
[tex]\[
2 \cdot x = -0.8
\][/tex]
Solving for [tex]\(x\)[/tex],
[tex]\[
x = \frac{-0.8}{2} = -0.4
\][/tex]
So, the number by which [tex]\(\left(\frac{1}{2}\right)^{-1}\)[/tex] should be multiplied so that the product is [tex]\(\left(\frac{-5}{4}\right)^{-1}\)[/tex] is [tex]\(-0.4\)[/tex].
