To show that a line is a representation of an infinite number of points, I like to give my algebra 1 classes an equation, like y = 2x + 3, and then give each student a couple of different ordered pairs - some that are solutions and some that aren't. I have them each work out their points, and then go to the board to plot an open or closed circle, depending. Once all the students sit back down, we look for patterns and see that all the closed circles fell on a straight line. Discuss, and voila.
This extends nicely to linear inequalities (and systems of equations and inequalities). On Tuesday, my algebra 2 students were reviewing linear inequalities so I did this activity with them. I really like it, because it is engaging, and it helps build a mental picture that they can rely on later on when they are struggling through graphing problems on their own. My students often get stuck on the "pick a test point" part of the process; but now, I ask them if they would have plotted a closed or open circle based on their result, and to think about what the picture on the board looked like. This usually helps them see which side of the boundary line to shade, and to be able to explain why.
Here is what the board looks like after students plotted their points:
Then, we looked for patterns. Usually, a student will come to the board and draw some sort of line after getting frustrated with trying to explain it in words. Then I reveal the shading: