Equations and Inequalities with No Solution or Infinite Solutions

I was looking for something a little bit different than what I had done in the past to introduce equations that have no solution or infinite solutions.  I came across this post from Sarah who blogs at Everybody is a Genius, and it was exactly what I was looking for.  I also liked this because when I had these students as 6th graders, I used scales to introduce them to solving equations, so this wasn’t a new idea for them.

I gave this sheet to students and told them to fill in the boxes to keep the scales balanced, and that for each scale, the number in the box must be the same.  Students have done a few different Open Middle problems this year, so some students struggled with the idea that they could no reuse numbers since they are used to not being able to reuse them for those problems, but they eventually understood what to do.

Equation Scales

As I was walking around, exactly what I hoped would happen, happened.  Students got two number 3 and I heard, “What?  This doesn’t make sense.”  “This is impossible.”

As we went over what students came up with, we discussed how in #1 and #4, we could pick any number we wanted, in #2 and #5 only one number works, and in #3, and #6 no numbers work.  Then we took some notes on this.  In the notes sheet I handed out to students, I included a picture of the scale and we wrote out the equation and showed what was happening to the scale as we did the algebra.

I liked that introducing this topic this way to students gave students a visual to help them understand these types of equations.


The next day we did some practice at the whiteboards.  I always include some problems that have one solution (especially ones where x = 0) because some students want to start saying every single equation either has no solution or infinite solutions, even though I stress that this only happens when the variables are eliminated.

Sarah Carter has created a nice Open Middle style problem to go with this topic.  Here students can use the numbers -4 to 4.

Screen Shot 2018-10-28 at 1.19.12 PM


Last week we worked on solving inequalities with infinite solutions and no solution.  I really liked what I did last year for this, so I did something similar this year.  I started the day by having students solve an equation that had no solution.  Then, I asked students which inequalities would make that true and which would make it false.

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We briefly discussed which would make it true and which would make it false, and that was pretty much the only instruction I gave students that day.  They had little to no trouble transferring the idea of equations with no solutions or infinite solutions to inequalities.

I shared at the end of this post a Desmos card sort I use as well as another Open Middle style problem on this topic.

screen-shot-2017-11-19-at-5-26-34-pmscreen-shot-2017-11-19-at-5-31-35-pm


Overall, I’m really happy with how students are doing with these types of problems.  I think that introducing this idea using the scales really helped my students to see what was going on.

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