Math 20-1 Chapter 9 Linear and Quadratic Inequalities 9.1 Linear Inequalities in Two Variables Teacher Notes

A linear inequality describes a region of a coordinate plane called a half-plane. All points in the region are solutions of the linear inequality. The boundary line of the region is the graph of the related equation. 9.1 Linear Inequalities in Two Variables Relational expressions of the type x + 2y ≤ 8 and 3x – y > 6 are called linear inequalities in two variables. A solution of a linear inequality in two variables is an ordered pair (x, y) which makes the inequality true. Example: (1, 3) is a solution to x + 2y ≤ 8 (1) + 2(3) ≤ 8 7 ≤ 8 TRUE 9.1.1

An ordered pair (x,y) is a solution if it makes the inequality true. Determine if the following points are solutions to the inequality: 3x + 2y ≥ 2 a) (0,0)b) (2,-1) c) (0,2) Linear Inequalities in Two Variables 3(0) + 2(0) ≥ 2 0 ≥ 2 Not a solution 3(2) + 2(-1) ≥ 2 4 ≥ 2 Is a solution 3(0) + 2(2) ≥ 2 4 ≥ 2 Is a solution 9.1.2

The boundary line separates the Cartesian plane into two regions. The boundary line is the related linear equality. Linear Inequalities in Two Variables When the inequality is written as y ≤ or y ≥, the points on the boundary line are solutions of the inequality and the line is solid. When the inequality is written as y, the points on the boundary line are not solutions of the inequality and the line is dashed

The solution set, of a linear inequality in two variables is the set of all solutions. The solution region is the set of all points in the Cartesian plane that satisfy an inequality. Linear Inequalities in Two Variables When the inequality is written as y > or y ≥, the points above the boundary line are solutions of the inequality. When the inequality is written as y < or y ≤, the points below the boundary line are solutions of the inequality

Graph: 3x – 5y < 15. Graph the related linear equation. 3x – 5y = 15. 3x – 5y = 15 3x – 5(0) = 15 3x = 15 x = 5 3x – 5y = 15 3(0) – 5y = 15 –5y = 15 y = –3 x-intercept: y= 0y-intercept: x = 0 Use intercepts to graph this line. Graphing Linear Inequalities in Two Variables 3x – 5y = 15 Why is the boundary line a broken line? 9.1.5

Choose a test point that is not on the line. Graphing Linear Inequalities in Two Variables 3x – 5y = 15 Choose (0, 0) (0, 0) If a true statement results, shade the half- plane containing the test point. The points above the line of 3x – 5y = 15 satisfy the inequality 3x – 5y < 15. 3x – 5y < 15 3(0) – 5(0) < 15 0 – 0 < 15 0 < 15 TRUE Graph: 3x – 5y < 15. Verify that a point within the solution region satisfies the inequality. 3x – 5y < 15 3(–2) – 5(6) < 15 –6 – 30 < 15 –36 < 15 TRUE Choose (–2, 6) 9.1.6

Your Turn Graph the linear inequality 2x + y > 2 –Graph the associated equation 2x + y = 2 Draw a dashed line since the inequality is > –Test a point that is not on the boundary line Choose (0, 0) 2x + y > 2 2(0) + 0 > 2 0 > 2 False –Shade the half-plane that contains solutions to the inequality The test point is false. Therefore shade the half-plane that does not contain the point (0, 0) x + y =

Write an inequality to represent the graph. Write the equation in slope y-intercept form The graph is shaded above a solid boundary line. y-intercept: –3 slope: –2 Replace = with ≥ to write the inequality y ≥ –2x – 3. y = mx + b y = –2x –

Your Turn Write an inequality to represent the graph. y-intercept: 1; slope: Write an equation in slope- intercept form. The graph is shaded above a dashed boundary line. Replace = with > to write the inequality 9.1.9

You can spend at most $12.00 for refreshments for the friendship picnic. Iced tea costs $1.50 a gallon, and lemonade costs $2.00 per gallon. a)Write an inequality to describe the situation. b)Graph the solutions. c)Describe reasonable solutions. d)Give two possible combinations of drinks you could buy. Write and Solve an Inequality

1.50x y ≤ c) Only whole number solutions are reasonable. Iced Tea (gal) Lemonade (gal) d) Possible combination of refreshments: Iced Tea (gal) 242 Lemonade (gal) 314 Cost $9.00$8.00$11.00 Write and Solve an Inequality a)Let x represent the number of gallons of iced tea. Let y represent the number of gallons of lemonade b)

Suggested Questions: Page 472: 1d, 3c,d, 4b,e, 5a,b, 7, 8a, 9, 11, 13, 15