Answer
$\left(-\infty,\dfrac{5}{7} \right]$
Work Step by Step
$\bf{\text{Solution Outline:}}$
The domain of the given function, $
d(x)=-\sqrt[4]{5-7x}
,$ is the permissible values of $x.$
$\bf{\text{Solution Details:}}$
With an even index (index equals $
4
$), then the radicand should be a nonnegative number. Hence,
\begin{array}{l}\require{cancel}
5-7x\ge0
\\\\
-7x\ge-5
.\end{array}
Dividing both sides by a negative number (and consequently reversing the inequality symbol) the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-7x\ge-5
\\\\
x\le\dfrac{-5}{-7}
\\\\
x\le\dfrac{5}{7}
.\end{array}
Hence, the domain is the interval $
\left(-\infty,\dfrac{5}{7} \right]
.$