A possible solution

Here is one possible solution.

No real solutions Solutions lie between \(1\) and \(6\) Negative solutions only All values of \(x\) are solutions
Equations in \(x\) \[x^2 +4 =0\] \[5x - 3 = 9-x \] \[2x - 5 = -12\] \[(x-1)^2 -1 = x^2 - 2x\]
Inequalities \[x^2<0\] \[5 < 2x - 3 < 15\] \[8 -x < 2 - 3x\] \[x^2 ≥ 0\]
Simultaneous equations in \(x\) and \(y\) \(y = 3x\) and \(\frac{y}{3} - 1 = x\) \(3y = x\) and \(y = x - 2\) \(y = 3x+5\) and \(y = -x-3\) \(y = 2x\) and \(\frac{y}{2}= x\)

Did you have any choice about the row and column headings?

We have put the first row as ‘Equations in \(x\)’, so examples all have a single variable only. How is this different from just writing ‘Equations’? Could simultaneous equations appear here if this was your row heading?

Can you simplify any of your examples?

Instead of giving the equation \(5x - 3 = 9-x\), which has a solution between \(1\) and \(6\), we could have written \(x = 2\). This is still an equation, albeit a very simple one, that has the properties required. Is it possible to give all the examples in this form, so you can immediately see whether they have the attributes of their cell?

Do all the examples require some ‘solving’ to check they have the attributes of the cell? If not, can you make it so that they all do?

If you gave an answer such as \(x = \sqrt{-4}\) for an equation that has no real solutions, you can easily see that it matches the properties of the cell. However, you could have written it as \(x^2 + 4 = 0\), so that it requires some manipulation to check it is correct.

If we required all the cells to contain quadratics, would it still be possible to fill all the cells?

It might be best to try to sketch what the answers might look like graphically to help you decide if they are all possible.