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txmedteach eNotes educator| Certified Educator

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


`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:


Let's subtract both sides by 1:


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


There you go! our solutions to both inequalities are:



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


I hope that helps!

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