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2-7 Two-Variable Inequalities

2-7 Two-Variable Inequalities. M11.D.2.1.2: Identify or graph functions, linear equations, or linear inequalities on a coordinate plane. Objectives. Graphing Linear Inequalities Graphing Two-Variable Absolute Value Inequalties. Vocabulary.

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2-7 Two-Variable Inequalities

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  1. 2-7 Two-Variable Inequalities M11.D.2.1.2: Identify or graph functions, linear equations, or linear inequalities on a coordinate plane

  2. Objectives Graphing Linear Inequalities Graphing Two-Variable Absolute Value Inequalties

  3. Vocabulary A linear inequality is an inequality in two variables whose graph is a region of the coordinate plane that is bounded by a line. A dashed boundary line means the line is not part of the solution. ex. > or < A solid boundary line means the line is part of the solution. ex. ≥ or ≤

  4. Vocabulary To figure out whether to shade above or below the line, Solve first for y. If y is greater or greater than or equal to the line, you shade above. ex. y > 2x + 1 If y is less than or less than or equal to the line, you shade below. ex. y ≤ 2x + 1

  5. 3 2 Step 1: Graph the boundary line y = x + 1. Since the inequality is greater than, not greater than or equal to, use a dashed boundary line. Graphing a Linear Inequality 3 2 Graph y > x + 1. Step 2: Since the inequality is greater than, y-values must be more than those on the boundary line. Shade the region above the boundary line.

  6. < – Relate: number of eggs needed for x orders of scrambled eggs number of eggs needed for y orders of omelets is less than or equal to 15 plus Define: Let x = the number of orders for scrambled eggs. Let y = the number of orders for omelets. 15 Write: 2 x + 3 y 15 Real World Example A restaurant has only 15 eggs until more are delivered. An order of scrambled eggs requires 2 eggs. An omelet requires 3 eggs. Write an inequality to model all possible combinations of orders of scrambled eggs and omelets the restaurant can fill till more eggs arrive. Graph the inequality.

  7. when y = 0, 2x + 3(0) = 15 2x = 15 x = 15 2 Graph the points ( , 0) and (0, 5). Since the inequality is less than or equal to, use a solid boundary line. 15 2 Continued (continued) Step 1: Find two points on the boundary line. Use the points to graph the boundary line. when x = 0, 2(0) + 3y = 15 3y = 15 y = 5

  8. Continued (continued) Step 2: Since the inequality is less than, y-values must be less than those on the boundary line. Shade the region below the boundary line. All ordered pairs with whole-number coordinates in the shaded area and on the boundary line represent a combination of x orders of scrambled eggs and y orders of omelets that the restaurant could fill.

  9. > – Since the inequality is greater than or equal to, the boundary is solid and the shaded region is above the boundary. Graphing Absolute Value Inequalities Graph y |2x| – 3.

  10. < – 1 2 b. Boundary: y = – x + 3 The boundary is solid. The shaded region is below the boundary. This is the graph of y – x + 3. 1 2 Writing Inequalities Write an inequality for each graph. The boundary line is given. a. Boundary: y = | x – 2| – 1 The boundary line is dashed. The shaded region is above the boundary. This is the graph of y > |x – 2| – 1.

  11. Homework Pg 104 & 105 # 1,2,10,11,12,20,21

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