The gradient of the line joining points M(4, 3x) and N(-2x, 1) is 2/3. Find: (a) the value of x; (b) correct to three significant figures, |MN|.

The gradient $m$ of a line joining two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $m = \frac{y_2 - y_1}{x_2 - x_1}$.
Given points $M(4, 3x)$ and $N(-2x, 1)$, and the gradient $m = \frac{2}{3}$.
Let $(x_1, y_1) = (4, 3x)$ and $(x_2, y_2) = (-2x, 1)$.

Cross-multiply:

Add $9x$ to both sides:

Add 8 to both sides:

The value of $x$ is $\boxed{2.2}$.

Substitute $x = 2.2$ into the coordinates of $M$ and $N$:
$M(4, 3x) = M(4, 3 \times 2.2) = M(4, 6.6)$
$N(-2x, 1) = N(-2 \times 2.2, 1) = N(-4.4, 1)$

The distance $|MN|$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

The length of $|MN|$ correct to three significant figures is $\boxed{10.1}$.