Previously, I shared how I use a Google Jamboard to introduce 1 variable linear equalities with a focus on conceptual understanding. I use the same approach for the 2 variable version (example problem below).
A conceptual gap that typically arises is the students do not understand what the shading represents. This is what I am addressing from the start using a Jamboard. First, the focus is on understanding the inequality and identifying a single point that works (below).
The next step is for students to determine more points that are solutions for the inequality, with no equal to part. (below).
The equal to part is addressed separately (below).
The equal to and the greater parts previously addressed are combined together.
The inequality is will be expanded to include an operation (+ 2) with a focus on the equal to part first.
The greater than with no equal to is addressed.
Then the equal to and greater than are addressed sequential. The equal to results in dots in a straight line and in lieu of plotting all the points, a line is drawn (building on the intro to 1 variable inequalities). This is followed by the greater than part and shading in lieu of plotting all of the dots above. THIS is where they gain an understanding of what the aforementioned shading is.
Finally, the dashed line is addressed by showing, as was done with the 1 variable inequalities, that there is a cutoff point that is not part of the solution set so in lieu of plotting a bunch of open circles, a dashed line is drawn.