The problem states that the gradient of the line joining points $M(4, 3x)$ and $N(-2x, 1)$ is $\frac{2}{3}$. We need to find the value of $x$ and then find the length of the line segment $MN$ correct to three significant figures.

1. Problem Description

The problem states that the gradient of the line joining points $M(4, 3x)$ and $N(-2x, 1)$ is $\frac{2}{3}$. We need to find the value of $x$ and then find the length of the line segment $MN$ correct to three significant figures.

2. Solution Steps

(a) The gradient $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

In this case, $M(4, 3x)$ and $N(-2x, 1)$ are the two points, and the gradient $m = \frac{2}{3}$.

So we have:

$\frac{2}{3} = \frac{1 - 3x}{-2x - 4}$

Multiplying both sides by $3(-2x - 4)$, we get:

$2(-2x - 4) = 3(1 - 3x)$

$-4x - 8 = 3 - 9x$

$-4x + 9x = 3 + 8$

$5x = 11$

$x = \frac{11}{5} = 2.2$

(b) Now, we need to find the length of the line segment $MN$ when $x = 2.2$.

The coordinates of $M$ are $(4, 3x) = (4, 3(2.2)) = (4, 6.6)$.

The coordinates of $N$ are $(-2x, 1) = (-2(2.2), 1) = (-4.4, 1)$.

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

So, the length of $MN$ is:

$|MN| = \sqrt{(-4.4 - 4)^2 + (1 - 6.6)^2}$

$|MN| = \sqrt{(-8.4)^2 + (-5.6)^2}$

$|MN| = \sqrt{70.56 + 31.36}$

$|MN| = \sqrt{101.92}$

$|MN| \approx 10.095543$

We need to round this to three significant figures.

$|MN| \approx 10.1$

3. Final Answer

(a) $x = 2.2$

(b) $|MN| = 10.1$