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# Integrated 2 Section 6-4

## on Dec 01, 2009

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Write and Graph Linear Inequalities

Write and Graph Linear Inequalities

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## Integrated 2 Section 6-4Presentation Transcript

• SECTION 6-4 Write and Graph Linear Inequalities
• ESSENTIAL QUESTIONS How do you write linear inequalities in two variables? How do you graph linear inequalities in two variables on the coordinate plane? Where you’ll see this: Business, market research, inventory
• VOCABULARY 1. Open Half-plane: 2. Boundary: 3. Linear Inequality: 4. Solution to the Inequality:
• VOCABULARY 1. Open Half-plane: A dashed boundary line separates the plane 2. Boundary: 3. Linear Inequality: 4. Solution to the Inequality:
• VOCABULARY 1. Open Half-plane: A dashed boundary line separates the plane 2. Boundary: The line that separates half-planes 3. Linear Inequality: 4. Solution to the Inequality:
• VOCABULARY 1. Open Half-plane: A dashed boundary line separates the plane 2. Boundary: The line that separates half-planes 3. Linear Inequality: A sentence where instead of an = sign, we use <, >, ≤, ≥, or ≠ 4. Solution to the Inequality:
• VOCABULARY 1. Open Half-plane: A dashed boundary line separates the plane 2. Boundary: The line that separates half-planes 3. Linear Inequality: A sentence where instead of an = sign, we use <, >, ≤, ≥, or ≠ 4. Solution to the Inequality: ANY ordered pair that makes the inequality true
• VOCABULARY 5. Graph of the Inequality: 6. Closed Half-plane: 7.Test Point:
• VOCABULARY 5. Graph of the Inequality: Includes graphing the boundary line and the shaded half-plane that includes the solution 6. Closed Half-plane: 7.Test Point:
• VOCABULARY 5. Graph of the Inequality: Includes graphing the boundary line and the shaded half-plane that includes the solution 6. Closed Half-plane: A solid boundary line separates the plane 7.Test Point:
• VOCABULARY 5. Graph of the Inequality: Includes graphing the boundary line and the shaded half-plane that includes the solution 6. Closed Half-plane: A solid boundary line separates the plane 7.Test Point: A point NOT on the boundary line that is used to test whether to shade above or below the boundary line
• GRAPHING A LINEAR INEQUALITY
• GRAPHING A LINEAR INEQUALITY Begin by treating the inequality as an equation to graph the boundary line and isolate y.
• GRAPHING A LINEAR INEQUALITY Begin by treating the inequality as an equation to graph the boundary line and isolate y. If <, >, or ≠, the boundary line will be dashed.
• GRAPHING A LINEAR INEQUALITY Begin by treating the inequality as an equation to graph the boundary line and isolate y. If <, >, or ≠, the boundary line will be dashed. If ≤ or ≥, the boundary line will be solid.
• GRAPHING A LINEAR INEQUALITY Begin by treating the inequality as an equation to graph the boundary line and isolate y. If <, >, or ≠, the boundary line will be dashed. If ≤ or ≥, the boundary line will be solid. Use a test point to determine shading OR
• GRAPHING A LINEAR INEQUALITY Begin by treating the inequality as an equation to graph the boundary line and isolate y. If <, >, or ≠, the boundary line will be dashed. If ≤ or ≥, the boundary line will be solid. Use a test point to determine shading OR If y is isolated, < and ≤ shade below, > and ≥ shade above
• EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the bondary line solid or dashed? a. 2x − 3y < 0 (3, 5), (4, 0)
• EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the bondary line solid or dashed? a. 2x − 3y < 0 (3, 5), (4, 0) 2(3) − 3(5) < 0
• EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the bondary line solid or dashed? a. 2x − 3y < 0 (3, 5), (4, 0) 2(3) − 3(5) < 0 6 −15 < 0
• EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the bondary line solid or dashed? a. 2x − 3y < 0 (3, 5), (4, 0) 2(3) − 3(5) < 0 6 −15 < 0 −9 < 0
• EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the bondary line solid or dashed? a. 2x − 3y < 0 (3, 5), (4, 0) 2(3) − 3(5) < 0 6 −15 < 0 −9 < 0 (3, 5) is a solution
• EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the bondary line solid or dashed? a. 2x − 3y < 0 2(4) − 3(0) < 0 (3, 5), (4, 0) 2(3) − 3(5) < 0 6 −15 < 0 −9 < 0 (3, 5) is a solution
• EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the bondary line solid or dashed? a. 2x − 3y < 0 2(4) − 3(0) < 0 (3, 5), (4, 0) 8−0<0 2(3) − 3(5) < 0 6 −15 < 0 −9 < 0 (3, 5) is a solution
• EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the bondary line solid or dashed? a. 2x − 3y < 0 2(4) − 3(0) < 0 (3, 5), (4, 0) 8−0<0 2(3) − 3(5) < 0 8<0 6 −15 < 0 −9 < 0 (3, 5) is a solution
• EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the bondary line solid or dashed? a. 2x − 3y < 0 2(4) − 3(0) < 0 (3, 5), (4, 0) 8−0<0 2(3) − 3(5) < 0 8<0 6 −15 < 0 (4, 0) is not a solution −9 < 0 (3, 5) is a solution
• EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the bondary line solid or dashed? a. 2x − 3y < 0 2(4) − 3(0) < 0 (3, 5), (4, 0) 8−0<0 2(3) − 3(5) < 0 8<0 6 −15 < 0 (4, 0) is not a solution −9 < 0 The boundary line is dashed (3, 5) is a solution
• EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the bondary line solid or dashed? b. 4y − x ≥ −6 (-2, -6), (0, 0)
• EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the bondary line solid or dashed? b. 4y − x ≥ −6 (-2, -6), (0, 0) 4(−6) − (−2) ≥ −6
• EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the bondary line solid or dashed? b. 4y − x ≥ −6 (-2, -6), (0, 0) 4(−6) − (−2) ≥ −6 −24 + 2 ≥ −6
• EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the bondary line solid or dashed? b. 4y − x ≥ −6 (-2, -6), (0, 0) 4(−6) − (−2) ≥ −6 −24 + 2 ≥ −6 −22 ≥ −6
• EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the bondary line solid or dashed? b. 4y − x ≥ −6 (-2, -6), (0, 0) 4(−6) − (−2) ≥ −6 −24 + 2 ≥ −6 −22 ≥ −6 (-2, -6) is not a solution
• EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the bondary line solid or dashed? b. 4y − x ≥ −6 4(0) − 0 ≥ −6 (-2, -6), (0, 0) 4(−6) − (−2) ≥ −6 −24 + 2 ≥ −6 −22 ≥ −6 (-2, -6) is not a solution
• EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the bondary line solid or dashed? b. 4y − x ≥ −6 4(0) − 0 ≥ −6 (-2, -6), (0, 0) 0 − 0 ≥ −6 4(−6) − (−2) ≥ −6 −24 + 2 ≥ −6 −22 ≥ −6 (-2, -6) is not a solution
• EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the bondary line solid or dashed? b. 4y − x ≥ −6 4(0) − 0 ≥ −6 (-2, -6), (0, 0) 0 − 0 ≥ −6 4(−6) − (−2) ≥ −6 0 ≥ −6 −24 + 2 ≥ −6 −22 ≥ −6 (-2, -6) is not a solution
• EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the bondary line solid or dashed? b. 4y − x ≥ −6 4(0) − 0 ≥ −6 (-2, -6), (0, 0) 0 − 0 ≥ −6 4(−6) − (−2) ≥ −6 0 ≥ −6 −24 + 2 ≥ −6 (0, 0) is a solution −22 ≥ −6 (-2, -6) is not a solution
• EXAMPLE 1 Tell whether the given coordinates satisfy each inequality by testing each point. Is the bondary line solid or dashed? b. 4y − x ≥ −6 4(0) − 0 ≥ −6 (-2, -6), (0, 0) 0 − 0 ≥ −6 4(−6) − (−2) ≥ −6 0 ≥ −6 −24 + 2 ≥ −6 (0, 0) is a solution −22 ≥ −6 The boundary line is solid (-2, -6) is not a solution
• EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5
• EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5 m=3
• EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5 m = 3 Up 3, right 1
• EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5 m = 3 Up 3, right 1 y-int: (0, -5)
• EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5 m = 3 Up 3, right 1 y-int: (0, -5) Boundary line is dashed
• EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5 m = 3 Up 3, right 1 y-int: (0, -5) Boundary line is dashed
• EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5 m = 3 Up 3, right 1 y-int: (0, -5) Boundary line is dashed
• EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5 m = 3 Up 3, right 1 y-int: (0, -5) Boundary line is dashed
• EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5 m = 3 Up 3, right 1 y-int: (0, -5) Boundary line is dashed
• EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5 m = 3 Up 3, right 1 y-int: (0, -5) Boundary line is dashed
• EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5 m = 3 Up 3, right 1 y-int: (0, -5) Boundary line is dashed
• EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5 m = 3 Up 3, right 1 y-int: (0, -5) Boundary line is dashed Check (0, 0):
• EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5 m = 3 Up 3, right 1 y-int: (0, -5) Boundary line is dashed Check (0, 0): 0 > 3(0) − 5
• EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5 m = 3 Up 3, right 1 y-int: (0, -5) Boundary line is dashed Check (0, 0): 0 > 3(0) − 5
• EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5 m = 3 Up 3, right 1 y-int: (0, -5) Boundary line is dashed Check (0, 0): 0 > 3(0) − 5
• EXAMPLE 2 Graph the following inequalities. a. y > 3x − 5 m = 3 Up 3, right 1 y-int: (0, -5) Boundary line is dashed Check (0, 0): 0 > 3(0) − 5
• EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2
• EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3 m=− 2
• EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3 m = − Down 3, right 2 2
• EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3 m = − Down 3, right 2 2 y-int: (0, 4)
• EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3 m = − Down 3, right 2 2 y-int: (0, 4) Boundary line is solid
• EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3 m = − Down 3, right 2 2 y-int: (0, 4) Boundary line is solid
• EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3 m = − Down 3, right 2 2 y-int: (0, 4) Boundary line is solid
• EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3 m = − Down 3, right 2 2 y-int: (0, 4) Boundary line is solid
• EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3 m = − Down 3, right 2 2 y-int: (0, 4) Boundary line is solid
• EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3 m = − Down 3, right 2 2 y-int: (0, 4) Boundary line is solid
• EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3 m = − Down 3, right 2 2 y-int: (0, 4) Boundary line is solid
• EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3 m = − Down 3, right 2 2 y-int: (0, 4) Boundary line is solid Check (0, 0):
• EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3 m = − Down 3, right 2 2 y-int: (0, 4) Boundary line is solid 3 Check (0, 0): 0 ≤ − (0) + 4 2
• EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3 m = − Down 3, right 2 2 y-int: (0, 4) Boundary line is solid 3 Check (0, 0): 0 ≤ − (0) + 4 2
• EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3 m = − Down 3, right 2 2 y-int: (0, 4) Boundary line is solid 3 Check (0, 0): 0 ≤ − (0) + 4 2
• EXAMPLE 2 Graph the following inequalities. 3 b. y ≤ − x + 4 2 3 m = − Down 3, right 2 2 y-int: (0, 4) Boundary line is solid 3 Check (0, 0): 0 ≤ − (0) + 4 2
• WHERE TO SHADE When y is isolated, there is a trick we can use:
• WHERE TO SHADE When y is isolated, there is a trick we can use: y goes down when we get less (<, ≤), so shade below
• WHERE TO SHADE When y is isolated, there is a trick we can use: y goes down when we get less (<, ≤), so shade below y goes up when we get less (>, ≥), so shade above
• EXAMPLE 3 Rectangle ABCD has a perimeter of at least 10 cm. a. Write a linear inequality that represents the situation.
• EXAMPLE 3 Rectangle ABCD has a perimeter of at least 10 cm. a. Write a linear inequality that represents the situation. x = length, y = width
• EXAMPLE 3 Rectangle ABCD has a perimeter of at least 10 cm. a. Write a linear inequality that represents the situation. x = length, y = width P = 2x + 2y
• EXAMPLE 3 Rectangle ABCD has a perimeter of at least 10 cm. a. Write a linear inequality that represents the situation. x = length, y = width P = 2x + 2y 10 ≤ 2x + 2y
• EXAMPLE 3 Rectangle ABCD has a perimeter of at least 10 cm. a. Write a linear inequality that represents the situation. x = length, y = width P = 2x + 2y 10 ≤ 2x + 2y -2x -2x
• EXAMPLE 3 Rectangle ABCD has a perimeter of at least 10 cm. a. Write a linear inequality that represents the situation. x = length, y = width P = 2x + 2y 10 ≤ 2x + 2y -2x -2x 10 − 2x ≤ 2y
• EXAMPLE 3 Rectangle ABCD has a perimeter of at least 10 cm. a. Write a linear inequality that represents the situation. x = length, y = width P = 2x + 2y 10 ≤ 2x + 2y -2x -2x 10 − 2x ≤ 2y 2 2
• EXAMPLE 3 Rectangle ABCD has a perimeter of at least 10 cm. a. Write a linear inequality that represents the situation. x = length, y = width P = 2x + 2y 10 ≤ 2x + 2y -2x -2x 5− x ≤ y 10 − 2x ≤ 2y 2 2
• EXAMPLE 3 Rectangle ABCD has a perimeter of at least 10 cm. a. Write a linear inequality that represents the situation. x = length, y = width P = 2x + 2y 10 ≤ 2x + 2y -2x -2x 5− x ≤ y 10 − 2x ≤ 2y 2 2 y ≥ −x + 5
• EXAMPLE 3 b. Graph the solution to the inequality. y ≥ −x + 5
• EXAMPLE 3 b. Graph the solution to the inequality. y ≥ −x + 5
• EXAMPLE 3 b. Graph the solution to the inequality. y ≥ −x + 5
• EXAMPLE 3 b. Graph the solution to the inequality. y ≥ −x + 5
• EXAMPLE 3 b. Graph the solution to the inequality. y ≥ −x + 5
• EXAMPLE 3 b. Graph the solution to the inequality. y ≥ −x + 5
• EXAMPLE 3 b. Graph the solution to the inequality. y ≥ −x + 5
• EXAMPLE 3 b. Graph the solution to the inequality. y ≥ −x + 5
• EXAMPLE 3 b. Graph the solution to the inequality. y ≥ −x + 5
• EXAMPLE 3 b. Graph the solution to the inequality. y ≥ −x + 5
• EXAMPLE 3 b. Graph the solution to the inequality. y ≥ −x + 5
• EXAMPLE 3 c. Does the “trick” tell us to shade above or below the boundary line? How do you know? d. Use the graph to name three possible combinations of length and width for rectangle ABCD. Check to make sure they satisfy the situation.
• EXAMPLE 3 c. Does the “trick” tell us to shade above or below the boundary line? How do you know? You shade above, as y gets larger due to ≥ d. Use the graph to name three possible combinations of length and width for rectangle ABCD. Check to make sure they satisfy the situation.
• EXAMPLE 3 c. Does the “trick” tell us to shade above or below the boundary line? How do you know? You shade above, as y gets larger due to ≥ d. Use the graph to name three possible combinations of length and width for rectangle ABCD. Check to make sure they satisfy the situation. Any points on the line or the shaded region work. The values must be positive in this situation.
• HOMEWORK
• HOMEWORK p. 260 #1-37 odd “Everyone has talent. What is rare is the courage to follow the talent to the dark place where it leads.” - Erica Jong