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