To start, let's recognize that the double inequality is actually two distinct inequalities, which we'll write here:

`-7<=4x+1`

`4x+1 < 13`

So, let's solve them separately using techniques from algebra. The big deal here is that **when you multiply or divide both sides by a negative number, the inequality flips**. Also, nothing we do will ever switch a less-than/greater than to a less-than-or-equal-to/greater-than-or-equal-to. **The equal sign is either always there or always not there!** We'll need to watch out for this to make sure we don't get mixed up! Let's start with the first inequality:

`-7 <= 4x+1`

We'll start by subtracting 1 from both sides:

`-8 <=4x`

Now, we divide both sides by 4. Notice that we're dividing by a positive number, so we're not going to flip the inequality:

`-2 <= x`

Well, there's our first solved inequality!

Now, our second one:

`4x+1<13`

Let's subtract both sides by 1:

`4x<12`

Now, we divide both sides by 4 (again, a positive number, so no inequality-flipping):

`x<3`

There you go! our solutions to both inequalities are:

`-2<=x`

`x<3`

We can now recombine these into a form that looks like the problem!

`-2<=x<3`

I hope that helps!