-x - 17 < 4x - 2 < 7x - 2

This inequality is actually two inequalities. We can solve it step by step:

1. -x - 17 < 4x - 2
2. 4x - 2 < 7x - 2

Let's solve the first inequality:

-x - 17 < 4x - 2

Add x to both sides:

-17 < 5x - 2

Add 2 to both sides:

-15 < 5x

Divide by 5:

-3 < x

So, x > -3

Now, let's solve the second inequality:

4x - 2 < 7x - 2

Subtract 4x from both sides:

-2 < 3x - 2

Add 2 to both sides:

0 < 3x

Divide by 3:

0 < x

So, x > 0

The solution to the original inequality is the intersection of the solutions to these two inequalities. So, the solution is x > 0 and x > -3, which simplifies to x > 0.