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Write and Graph Linear Inequalities
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SECTION 6-4Write and Graph Linear Inequalities
Tue, Dec 01
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
Tue, Dec 01
VOCABULARY
1. Open Half-plane:
2. Boundary:
3. Linear Inequality:
4. Solution to the Inequality:
Tue, Dec 01
VOCABULARY
1. Open Half-plane: A dashed boundary line separates the plane
2. Boundary:
3. Linear Inequality:
4. Solution to the Inequality:
Tue, Dec 01
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:
Tue, Dec 01
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:
Tue, Dec 01
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
Tue, Dec 01
VOCABULARY
5. Graph of the Inequality:
6. Closed Half-plane:
7. Test Point:
Tue, Dec 01
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:
Tue, Dec 01
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:
Tue, Dec 01
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
Tue, Dec 01
GRAPHING A LINEAR INEQUALITY
Tue, Dec 01
Begin by treating the inequality as an equation to graph the boundary line and isolate y.
GRAPHING A LINEAR INEQUALITY
Tue, Dec 01
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
Tue, Dec 01
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
Tue, Dec 01
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
Tue, Dec 01
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
GRAPHING A LINEAR INEQUALITY
Tue, Dec 01
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)
Tue, Dec 01
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
Tue, Dec 01
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
Tue, Dec 01
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
Tue, Dec 01
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
Tue, Dec 01
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
2(4)− 3(0) < 0
Tue, Dec 01
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
2(4)− 3(0) < 0
8 − 0 < 0
Tue, Dec 01
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
2(4)− 3(0) < 0
8 − 0 < 0 8 < 0
Tue, Dec 01
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
2(4)− 3(0) < 0
8 − 0 < 0 8 < 0
(4, 0) is not a solution
Tue, Dec 01
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
2(4)− 3(0) < 0
8 − 0 < 0 8 < 0
(4, 0) is not a solution
The boundary line is dashed
Tue, Dec 01
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)
Tue, Dec 01
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
Tue, Dec 01
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
Tue, Dec 01
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
Tue, Dec 01
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
Tue, Dec 01
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
4(0)− 0 ≥ −6
Tue, Dec 01
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
4(0)− 0 ≥ −6
0 − 0 ≥ −6
Tue, Dec 01
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
4(0)− 0 ≥ −6
0 − 0 ≥ −6 0 ≥ −6
Tue, Dec 01
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
4(0)− 0 ≥ −6
0 − 0 ≥ −6 0 ≥ −6
(0, 0) is a solution
Tue, Dec 01
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
4(0)− 0 ≥ −6
0 − 0 ≥ −6 0 ≥ −6
(0, 0) is a solution
The boundary line is solid
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3 Up 3, right 1
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3 Up 3, right 1
y-int: (0, -5)
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3 Up 3, right 1
y-int: (0, -5)
Boundary line is dashed
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3 Up 3, right 1
y-int: (0, -5)
Boundary line is dashed
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3 Up 3, right 1
y-int: (0, -5)
Boundary line is dashed
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3 Up 3, right 1
y-int: (0, -5)
Boundary line is dashed
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3 Up 3, right 1
y-int: (0, -5)
Boundary line is dashed
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3 Up 3, right 1
y-int: (0, -5)
Boundary line is dashed
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
a. y > 3x − 5
m = 3 Up 3, right 1
y-int: (0, -5)
Boundary line is dashed
Tue, Dec 01
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):
Tue, Dec 01
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
Tue, Dec 01
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
Tue, Dec 01
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
Tue, Dec 01
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
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
b. y ≤ −
32
x + 4
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
b. y ≤ −
32
x + 4
m = −
32
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
b. y ≤ −
32
x + 4
Down 3, right 2 m = −
32
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
b. y ≤ −
32
x + 4
Down 3, right 2
y-int: (0, 4) m = −
32
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
b. y ≤ −
32
x + 4
Down 3, right 2
y-int: (0, 4)
Boundary line is solid
m = −
32
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
b. y ≤ −
32
x + 4
Down 3, right 2
y-int: (0, 4)
Boundary line is solid
m = −
32
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
b. y ≤ −
32
x + 4
Down 3, right 2
y-int: (0, 4)
Boundary line is solid
m = −
32
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
b. y ≤ −
32
x + 4
Down 3, right 2
y-int: (0, 4)
Boundary line is solid
m = −
32
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
b. y ≤ −
32
x + 4
Down 3, right 2
y-int: (0, 4)
Boundary line is solid
m = −
32
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
b. y ≤ −
32
x + 4
Down 3, right 2
y-int: (0, 4)
Boundary line is solid
m = −
32
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
b. y ≤ −
32
x + 4
Down 3, right 2
y-int: (0, 4)
Boundary line is solid
m = −
32
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
b. y ≤ −
32
x + 4
Down 3, right 2
y-int: (0, 4)
Boundary line is solid
Check (0, 0):
m = −
32
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
b. y ≤ −
32
x + 4
Down 3, right 2
y-int: (0, 4)
Boundary line is solid
Check (0, 0):
m = −
32
0 ≤ −
32
(0)+ 4
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
b. y ≤ −
32
x + 4
Down 3, right 2
y-int: (0, 4)
Boundary line is solid
Check (0, 0):
m = −
32
0 ≤ −
32
(0)+ 4
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
b. y ≤ −
32
x + 4
Down 3, right 2
y-int: (0, 4)
Boundary line is solid
Check (0, 0):
m = −
32
0 ≤ −
32
(0)+ 4
Tue, Dec 01
EXAMPLE 2
Graph the following inequalities.
b. y ≤ −
32
x + 4
Down 3, right 2
y-int: (0, 4)
Boundary line is solid
Check (0, 0):
m = −
32
0 ≤ −
32
(0)+ 4
Tue, Dec 01
WHERE TO SHADE
Tue, Dec 01
WHERE TO SHADE
When y is isolated, there is a trick we can use:
Tue, Dec 01
WHERE TO SHADE
When y is isolated, there is a trick we can use:
y goes down when we get less (<, ≤), so shade below
Tue, Dec 01
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
Tue, Dec 01
EXAMPLE 3
Rectangle ABCD has a perimeter of at least 10 cm.
a. Write a linear inequality that represents the situation.
Tue, Dec 01
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
Tue, Dec 01
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
Tue, Dec 01
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
Tue, Dec 01
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
Tue, Dec 01
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
Tue, Dec 01
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 ≤ 2y22
Tue, Dec 01
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 ≤ 2y22
5 − x ≤ y
Tue, Dec 01
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 ≤ 2y22
5 − x ≤ y
y ≥ −x + 5
Tue, Dec 01
EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5
Tue, Dec 01
EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5
Tue, Dec 01
EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5
Tue, Dec 01
EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5
Tue, Dec 01
EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5
Tue, Dec 01
EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5
Tue, Dec 01
EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5
Tue, Dec 01
EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5
Tue, Dec 01
EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5
Tue, Dec 01
EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5
Tue, Dec 01
EXAMPLE 3
b. Graph the solution to the inequality.
y ≥ −x + 5
Tue, Dec 01
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.
Tue, Dec 01
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.
Tue, Dec 01
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.
Tue, Dec 01
HOMEWORK
Tue, Dec 01
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
Tue, Dec 01