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**Graphing inequalities in two variables**

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**Inequalities in two variables**

Linear inequalities can be given in two variables x and y. For example, x + y > 3 Can you name a pair of values that satisfy the inequality? Some example solution pairs are: x = 1 and y = 4 x = –1 and y = 5 x = 4 and y = 5 x = 12 and y = –6 These 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.

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**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. y The region where x + y > 3 does not include points where x + y = 3 and so we draw this as a dotted line. 5 4 3 2 1 x –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 –1 We need to decide which part of the graph is the solution set. How can we check? –2 Mathematical Practices 7) 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

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Checking the origin Choose a point off the line and see if it satisfies the inequality. It is often easiest to check the origin, (0, 0). Is the origin in the region x + y > 3? Substitute x = 0 and y = 0 into the inequality x + y > 3: y 5 4 x + y > 3 3 2 0 + 0 > 3 1 x 0 > 2 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 –1 Teacher notes We can see that substituting 0 for x and y can be done quickly mentally. The point (0, 0) is also easy to locate. Mathematical Practices 7) Look for and make use of structure. Students should see that substituting (0, 0) into the inequality produces an inequality that is not true, so the origin must not lie within the solution region. They should be able to conclude that the solution region is therefore above the line. Image credit: © Alex Kalmbach, Shutterstock.com 2012 –2 0 is not greater than 2, so the origin is not in the solution region. x + y < 3 –3 –4 –5 The region representing x + y > 3 is therefore the region above the line.

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**Inequalities in two variables**

y Represent the solutions to the inequality 2y – x < 4 on a graph. 5 4 3 2 1 x Draw the line 2y – x = 4. –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 –1 –2 Check if the origin is in the region by substituting x = 0 and y = 0 into the inequality 2y – x < 4: 2y – x < 4 –3 –4 –5 This is true, so we know that (0, 0) is in the solution set. 2(0) – 0 = 0 < 4 Mathematical Practices 7) Look for and make use of structure. Students should see that substituting (0, 0) into the inequality produces an inequality that is true, so the origin must lie within the solution region. They should be able to conclude that the solution region is therefore below the line. Image credit © BruceParrott, Shutterstock.com 2012 The solution set is therefore the region below the line.

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Cars and trucks A ferry cannot hold more than 30 tons. If it holds x cars weighing 1 ton each and y trucks weighing 3 tons each, write down an inequality in x and y. x + 3y ≤ 30 If 20 cars were already on board, how many more trucks could the ferry carry? Substitute into x + 3y ≤ 30 and solve for y, Teacher notes Point out that we cannot have a fraction of a truck, so the maximum number of trucks that would fit on the ferry with 20 cars is 3 trucks. Mathematical Practices 2) Reason abstractly and quantitatively. Students should be able to decontextualize – take the given situation and represent it as an inequality. They should also be able to contextualize, i.e. realize that the number of trucks (y) must be a positive integer. They should take this into account by rounding down their final answer. 4) Model with mathematics. This slide demonstrates how students should be able to apply their knowledge of inequalities to solve problems arising in everyday life. They should be able to write an inequality to represent the situation and use it to solve a problem in context. Photo credit: © Panos Karapanagiotis, Shutterstock.com 2012 20 + 3y ≤ 30 subtract 20 from both sides: 3y ≤ 10 divide both sides by 3: y ≤ 3.3 (to nearest tenth) The ferry can hold 3 more trucks.

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Cars and trucks Show 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. y 10 x + 3y = 30 Teacher notes Going 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 Practices 2) 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. x 10 20 30 The points on the graph represent the solution set.

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**Uncles and aunts “Two of my aunts live together.”**

“Even if all my uncles and aunts brought a friend each, they still wouldn’t have enough people for a soccer team.” “I have more uncles than aunts.” Teacher notes In order to use the graphing tool, the inequalities must use the symbols ≤ or ≥. Some inequalities will need to be converted into this form. Let the number of uncles be x and the number of aunts be y. From the first statement we conclude: y ≥ 2 From the second statement we conclude: x > y so x – y ≥ 1 (ready for the graphing tool). From the third statement we conclude: 2(x + y) < 11 so 2x + 2y ≤ 10 (ready for the graphing tool). Draw a graph to find out what possibilities there are for x and y. This can be done using the flash activity on the following slide; conclude that the only possibility is that x = 3 uncles and y = 2 aunts. Mathematical Practices 2) Reason abstractly and quantitatively. Students should be able to decontextualize – take the given situation and represent it as a set of inequalities. They should be able to manipulate the inequalities in order to produce equivalent inequalities which can be applied to the graphing tool on the next slide, e.g. x > y so x – y ≥ 1. 4) Model with mathematics. This slide demonstrates how students should be able to apply their knowledge of inequalities to solve problems arising in everyday life. They should be able to write an inequality to represent the situation and use it to solve a problem in context. Photo credit: © Julie Keen, Shutterstock.com 2012 How many uncles and aunts does the girl have? Use the inequality graphing tool on the next slide to confirm your answer.

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