Graph each inequality on a number line.1. x > –5 2. y ≤ 0

3. Write –6x + 2y = –4 in slope-intercept form, and graph.

y = 3x – 2

Warm-Up

Inequalities

Learning Targets

Review Inequalities

Graphing Inequalities

Knowing where Possible Solutions Exist

Testing for Possible Solutions

What is an Inequality?

Definition: two expressions are not equal to one another.

Equality vs. Inequality

Equality sets expressions values so that they are equal to one another.Ex:

Inequality compares the two expressions values.Ex:

Number Lines

Equalities on a number line:

Inequalities on a number line:

-1 0 1

-1 0 1

-1 0 1

-1 0 1 -1 0 1

Graphically

Equalities on a coordinate plane:

Inequalities on a coordinate plane:

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9-8-7-6-5-4-3-2-1

123456789

x

y

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9-8-7-6-5-4-3-2-1

123456789

x

y

𝑦=2 𝑥+3 𝑦 ≤ 2 𝑥+3

Boundary Line

Different Symbols… Different Lines/Points

≤ , ≥

¿ ,>¿

Solid Line, Filled Point

Dashed Line, Hollow Point

Objective - To graph linear inequalities in the coordinate plane.

Number Line Coordinate Plane

-4 -3 -2 -1 0 1 2 3 4

x 3 y

x

x 3

x = 3

Graph y 2.

Number Line Coordinate Plane

-4 -3 -2 -1 0 1 2 3 4

y 2 y

x

y 2

y = -2

2Graph y x 1.3

Boundary Line2y x 13

2m3

b 1

Test a Point2y x 13

20 (0) 13

0 1 False!

y

x

Graph 4x 5y 10.

y

x

4x 5y 10 4x 4x

5y 4x 10 5 5

4y x 25

4m5

b 2

Solid line

If y = mx + b,

45

Graph 3x 2y 8. 3x 2y 8 3x 3x

2y 3x 8 2 2

Why do we flip the sign?Pause and Ponder…

Graph 3x 2y 8.

y

x

3x 2y 8 3x 3x

2y 3x 8 2 2

3y x 42

3m2

b 4

Dashed line

If y = mx + b,

32

Graph 4x 3y 6.

y

x

4x 3y 6 4x 4x

3y 4x 6 3 3

4y x 23

4m3

b 2

Solid line

If y = mx + b,

43

y

x

4m3

b 2

Dashed line

If y = mx + b,

Write the inequality described by the graph below.

4y x 23

-4

+3

Determine whether the given point is a solution to the inequality -2x + 3y < 9.

1) (2, -3) 2x 3y 9 (x, y)

2( ) 3 3( ) 92 4 9 9

13 9 Yes, (2,-3) is a solution.

2) (3, 5) 2x 3y 9 2( ) 33 ( ) 95

6 15 9 9 9

No, (3,5) is not a solution.

ProblemIf you have less than $5.00 in nickels and dimes, find an inequality and sketch a graph to describe how many of each coin you have.

Let n = # of nickelsLet d = # of dimes

0.05 n + 0.10 d < 5.00

or

5 n + 10 d < 500

5n + 10d < 500n d

0 50

100 0

0 10 20 30 40 50 60 70 80 90 100n

d

60

50

40

30

20

10

0

𝑦<−3 𝑥+10 𝑦 ≤ 𝑥+1

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11

-11-10

-9-8-7-6-5-4-3-2-1

123456789

1011

x

y

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11

-11-10

-9-8-7-6-5-4-3-2-1

123456789

1011

x

y

Systems of Inequalities

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11

-11-10

-9-8-7-6-5-4-3-2-1

123456789

1011

x

y

Solution:

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11

-11-10-9-8-7-6-5-4-3-2-1

123456789

1011

x

y

Graph both inequalities over

each other:

Therefore our answer lies in the combined shaded

region:

Non linear Inequalities

Non Linear Inequalities

In chapter 4 we studied parent graphs and their transformation.

In the next few slides you will be given non linear graphs. Use your knowledge of parent graphs to display a graph for each function.

Then use a test point to graph the inequalities that follow.

Graph:

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9-8-7-6-5-4-3-2-1

123456789

x

y

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9-8-7-6-5-4-3-2-1

123456789

x

y

Graph:

Graph:

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9-8-7-6-5-4-3-2-1

123456789

x

y

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9-8-7-6-5-4-3-2-1

123456789

x

y

Graph

Graph

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9-8-7-6-5-4-3-2-1

123456789

x

y

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9-8-7-6-5-4-3-2-1

123456789

x

y

Graph

Graph:

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9-8-7-6-5-4-3-2-1

123456789

x

y

Graph:

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9-8-7-6-5-4-3-2-1

123456789

x

y

Graph:

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9-8-7-6-5-4-3-2-1

123456789

x

y

Graph:

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9-8-7-6-5-4-3-2-1

123456789

x

y

Graph:

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9-8-7-6-5-4-3-2-1

123456789

x

y

Graph:

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9-8-7-6-5-4-3-2-1

123456789

x

y

On your own:

Review your notes. Rewrite and fortify them if needed.

Update your vocab list, if needed.