The problem asks us to express the solution to the compound inequality $233 + 6x \le 8x - 75 \le 475 + 6x$ in set notation.

1. Problem Description

The problem asks us to express the solution to the compound inequality $233 + 6x \le 8x - 75 \le 475 + 6x$ in set notation.

2. Solution Steps

We have the compound inequality $233 + 6x \le 8x - 75 \le 475 + 6x$. This inequality is equivalent to the two inequalities:

$233 + 6x \le 8x - 75$ and $8x - 75 \le 475 + 6x$

First, let's solve $233 + 6x \le 8x - 75$:

$233 + 6x \le 8x - 75$

Add 75 to both sides:

$233 + 75 + 6x \le 8x$

$308 + 6x \le 8x$

Subtract $6x$ from both sides:

$308 \le 8x - 6x$

$308 \le 2x$

Divide both sides by 2:

$154 \le x$

or $x \ge 154$

Next, let's solve $8x - 75 \le 475 + 6x$:

$8x - 75 \le 475 + 6x$

Subtract $6x$ from both sides:

$8x - 6x - 75 \le 475$

$2x - 75 \le 475$

Add 75 to both sides:

$2x \le 475 + 75$

$2x \le 550$

Divide both sides by 2:

$x \le 275$

Combining the two inequalities $x \ge 154$ and $x \le 275$, we have $154 \le x \le 275$.

3. Final Answer

$154 \le x \le 275$