We are asked to solve the inequality $\frac{1}{3}(5 - 3x) < \frac{2}{5}(3 - 7x)$ for $x$. Also, we are asked to express $m$ in terms of $k$ and $y$ in the equation $k = \sqrt{\frac{m-y}{m+1}}$.

1. Problem Description

We are asked to solve the inequality $\frac{1}{3}(5 - 3x) < \frac{2}{5}(3 - 7x)$ for $x$.

Also, we are asked to express $m$ in terms of $k$ and $y$ in the equation $k = \sqrt{\frac{m-y}{m+1}}$.

2. Solution Steps

First, we solve the inequality:

$\frac{1}{3}(5 - 3x) < \frac{2}{5}(3 - 7x)$

Multiply both sides by 15 to eliminate the fractions:

$15 \cdot \frac{1}{3}(5 - 3x) < 15 \cdot \frac{2}{5}(3 - 7x)$

$5(5 - 3x) < 6(3 - 7x)$

$25 - 15x < 18 - 42x$

Add $42x$ to both sides:

$25 - 15x + 42x < 18 - 42x + 42x$

$25 + 27x < 18$

Subtract 25 from both sides:

$25 + 27x - 25 < 18 - 25$

$27x < -7$

Divide both sides by 27:

$x < \frac{-7}{27}$

Now, we solve for $m$ in the equation $k = \sqrt{\frac{m-y}{m+1}}$

First, square both sides:

$k^2 = \frac{m-y}{m+1}$

Multiply both sides by $(m+1)$:

$k^2(m+1) = m-y$

$k^2m + k^2 = m - y$

Move terms with $m$ to one side and other terms to the other side:

$k^2m - m = -y - k^2$

Factor out $m$ on the left side:

$m(k^2 - 1) = -y - k^2$

Multiply both sides by $-1$:

$m(1 - k^2) = y + k^2$

Divide both sides by $(1 - k^2)$:

$m = \frac{y + k^2}{1 - k^2}$

3. Final Answer

For the inequality, the solution is $x < -\frac{7}{27}$. Therefore, the correct answer is D.

For making $m$ the subject of the relation, the solution is $m = \frac{y + k^2}{1 - k^2}$. Therefore, the correct answer is B.