2 Inequalities in two variables Linear inequalities can be given in two variables x and y.For example,x + y > 3Can you name a pair of values that satisfy the inequality?Some example solution pairs are:x = 1andy = 4x = –1andy = 5x = 4andy = 5x = 12andy = –6These solutions are usually written as coordinate pairs as(1, 4), (–1, 5), (4, 5), (12, –6).The whole solution set can be represented using a graph.
3 Graphing the inequality We can represent all the points where the x-coordinate and the y-coordinate add up to 3 with the line x + y = 3.yThe region where x + y > 3 does not include points where x + y = 3 and so we draw this as a dotted line.54321x–6–5–4–3–2–1123456–1We need to decide which part of the graph is the solution set.How can we check?–2Mathematical Practices7) Look for and make use of structure.Students should see that if they choose a point on either side of the line and substitute it into the inequality, they will see whether or not the point satisfies the inequality. This way they will be able to figure out which side of the line is the solution set.–3–4–5
7 Cars and trucksShow the possible numbers of cars and trucks that the ferry can carry on a graph.Start by drawing the x-axis between 0 and 30 and y-axis between 0 and 10.Next, draw the line x + 3y = 30.This point represents 20 cars and 3 trucks.y10x + 3y = 30Teacher notesGoing back to the original problem, we can see that the maximum number of cars that would fit on the ferry is 30, and the maximum number of trucks is 10. Since every solution will be within this range we can draw the x- and y- axes accordingly.The solution set is represented by a finite set of points rather than a region. This is because the numbers of cars or trucks can only be whole positive numbers.This is often the case for real life problems where discrete data is involved.Mathematical Practices2) Reason abstractly and quantitatively.Students should be able to decontextualize – take the given situation and represent it graphically. They should realize that the number of trucks and cars can only be whole numbers, so the graph is made up of single points plotted at each integer pair below the line.4) Model with mathematics.This slide demonstrates how students should be able to graph inequalities representing real-life problems.x102030The points on the graph represent the solution set.