Lesson 2-5: Graphing Linear Inequalities in Two Variables

Transcription

1 Inequalities on a graph Last chapter we worked with inequalities on a number line. We used a number line because the equations only had one variable. Today we will extend that thinking a bit from the number line to the x-y graph by working with inequalities with two variables. To get ourselves going, work through the following activities: 1. Graph the line x = 3. Next, shade in the graph showing everywhere that x > 3. You should end up with a graph that looks like this (graph on page 4 or follow the link). 2. Graph the line y = -5. Next, shade in the graph showing everywhere that y < -5. You should end up with a graph that looks like this. (graph on page 5 ) For both of these graphs, you shaded everything on the one side of the line. What you actually did was graph the inequalities x 3 (for the 1st) and y -5 (for the 2 nd ) if we include the graphed line itself. You shaded all the points (x, y) that are solutions for the inequalities. If that isn t clear, consider the first graph, the one for x 3. Pick a point that is in the shaded region; I ll pick (5, -2). The inequality is a greater than type so any point with an x coordinate bigger than (or equal) to 3 works since 5 is bigger than 3, this point should be in the shaded region. What about the point (1, 0)? Its x coordinate is less than 3, so it should not be in the shaded region. Page 1 of 5

2 What about slanted lines? Ok, the two examples above are pretty easy because they are horizontal and vertical lines. What about a line that is not horizontal or vertical? Graph the line y = x + 1. Now, before you start building your T-chart, what form is that equation in? It s in slope-intercept form. The slope is 1 (up 1, over 1) and the y-intercept is 1 or (0, 1). Use that information to quick graph the line: Now, what if I asked you to shade the region of the graph that is greater than the line? Would it be to the left or the right of the line? Before you rush to an answer, think about it. Any point that will be in the shaded region will have x and y values that satisfy the inequality. What is the inequality? It is y > x + 1. Take a look at the graph; pick a point that will be super easy to work with. The point (0, 0) would be pretty easy! Let s plug it into the inequality and see if it is a solution: y = 0 and x = 0 so we have 0 > or 0 > 1. Is that true? Err, no. Not in this time and space continuum! What does this mean? It means that (0, 0) is not a solution for y > x + 1 so it won t be in the shaded region. That means the shaded region must be on the other side (left side) of the line! And so it is! Not convinced? Try a point on the other side of the line from (0, 0). How about (-2, 3): y = 3 and x = -2 so we have 3 > or 3 > -1 which is true and in the shaded region! One more detail. Remember how we would plot < and on the number line? If the inequality included the equal bar underneath, we put a solid dot. If it didn t, we used an open/empty dot. We do something similar when graphing lines. If the inequality is < or > (doesn t include the equal bar) we mark the line with a dashed arrow line. If the inequality is or (does include the equal bar) we mark the line with a solid arrow line. Page 2 of 5

3 Our graph above is for the inequality y > x + 1 so should the line be dashed or solid? It should be dashed it does not have the equal bar underneath. Here s the graph: Graphing Inequalities Cookbook Here are the basics of how to graph linear inequalities: 1. Graph the line as an equality (example: for y < x + 1, graph y = x + 1) Use slope-intercept information to quick graph Dashed if > or < Solid if or 2. Test one point not on the line plug the x & y values into the inequality: If evaluates true, then the point is in the shaded region If evaluates false, then the point is not in the shaded region Remember: you only need to test one point! Page 3 of 5

4 Graph showing all the values for x greater than (or equal to) 3: Page 4 of 5

5 Graph showing all values for y less than (or equal to) -5. Page 5 of 5