After solving many different types of equations in 8th grade, inequalities are up next. We start by reviewing graphing inequalities before getting into solving them. Then we also work on inequalities that have all real numbers and no solution as answers.

##### Review of Graphing

Although students have seen inequalities and graphed them in the past, I’ve found that it is worth my time to spend a day or so giving students a quick refresher on this. There are several great Desmos activities for this. Here are a few that I’ve used and like.

- I use this one first.
- I also like this one.
- Students always enjoy Polygraphs.

##### Solving

In the past I had an activity I used to get students to discover when the inequality symbol needs to be switched when solving inequalities. It was sort of lengthy and cumbersome, but I didn’t know how to improve it more than I already had. Then I saw Sarah Tweet the picture below. It was EXACTLY what I was looking for! Thanks Sarah! Here is the link to download Sarah’s file.

Then for practice students do a Tarsia puzzle. I created the puzzle a while ago and don’t know where the file is that I can share. If you’re unfamiliar with Tarsia puzzles, you can learn more about them here.

I also have a question stack that I use for these types of problems. You can read about Sarah Carter’s question stacks here. You can download the file for the question stack I made, here.

##### All Real Numbers/No Solution

To introduce inequalities that have No Solution or All Real Numbers as the solution, I went back to what students already knew about equations like these. I had students solve a problem similar to the one below and then asked them what inequality symbol we could replace the equal sign with that would make the inequality have no solution and the same for all real numbers.

Then for practice, I had students work on** this **Desmos activity.

I also tried creating an Open Middle problem for these types of problems after seeing a similar one Sarah created for equations. I had one of my co-workers take a look at a different Open Middle problem I made, and he had a great idea from when he has used Open Middle problems in the past. He suggested to start by letting students use whatever numbers they want, and then after they come up with a solution to restrict them to only using certain numbers. I thought this was a great idea, so that’s what I did. I started by telling students they could use any integers they wanted as long as they didn’t repeat any of the 12 numbers. When a student came up with a solution, I said they could only use the integers -6 to 6.

You can download the files for the Open Middle puzzle **here**.