The problem is to rewrite the compound inequality $3x + 24 < 6(x+6) < 3x + 48$ as a double inequality in terms of $x$.

1. Problem Description

The problem is to rewrite the compound inequality $3x + 24 < 6(x+6) < 3x + 48$ as a double inequality in terms of $x$.

2. Solution Steps

First, simplify the middle term:

$6(x+6) = 6x + 36$

So the inequality becomes:

$3x + 24 < 6x + 36 < 3x + 48$

We have two inequalities:

$3x + 24 < 6x + 36$ and $6x + 36 < 3x + 48$

Let's solve the first inequality:

$3x + 24 < 6x + 36$

Subtract $3x$ from both sides:

$24 < 3x + 36$

Subtract $36$ from both sides:

$24 - 36 < 3x$

$-12 < 3x$

Divide by $3$:

$-4 < x$

$x > -4$

Now, let's solve the second inequality:

$6x + 36 < 3x + 48$

Subtract $3x$ from both sides:

$3x + 36 < 48$

Subtract $36$ from both sides:

$3x < 48 - 36$

$3x < 12$

Divide by $3$:

$x < 4$

Combining the two inequalities, we have $x > -4$ and $x < 4$.

This can be written as $-4 < x < 4$.

3. Final Answer

-4 < x < 4