Graphing Lines and Linear Inequalities

1
Graphing Lines and Linear Inequalities
Linear Inequalities
Linear Equations
Drive on The Education Highway
2
Linear Equations and Graphing 1. Parts of a
Coordinate Plane 2. Slope 3. Slope-intercept Form
of a Linear Equation 4. Graphing by x-
y-intercepts.
3
Parts of a coordinate plane.
4
Click on the correct quadrant numbers. Correct
answer applause.
6
5
Quadrant
Quadrant
4
I
II
III
IV
I
II
III
IV
3
2
1
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-1
Quadrant
Quadrant
-2
-3
I
II
III
IV
I
II
III
IV
-4
-5
-6
Lesson Start
5
Click on the correct axis names. Correct answer
clapping.
6
5
4
3
2
1
x-axis
y-axis
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-1
-2
-3
-4
-5
-6
x-axis
y-axis
Lesson Start
6
Click on the point for the origin. Correct
answer clapping.
6
5
4
3
2
1
x
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-1
-2
-3
-4
-5
-6
y
Lesson Start
7
Click on point (-3, 5). Correct point
applause.
6
5
4
3
2
1
x
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-1
-2
-3
-4
-5
-6
Lesson Start
y
8
You chose a line segment instead of a point. Go
back and try again.
INCORRECT ANSWER
9
You chose point (5, -3). Each ordered pair is in
the form (x, y) -- it follows the alphabet in its
internal order. You find the x value first, then
you find the y value. Where they meet is the
point. Go back and try again.
INCORRECT ANSWER
10
Click on the correct ordered pair for the black
point.
6
5
4
3
2
1
x
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-1
-2
-5/4
(-5, 4)
-3
(4, -5)
-4/5
-4
-5
-6
Lesson Start
y
11
You did not choose an ordered pair. Go back and
try again.
INCORRECT ANSWER
12
You chose the ordered pair for the pale green
point. Remember x comes before y in the
alphabet and in an ordered pair. Go back and try
again.
INCORRECT ANSWER
13
Return to Main Menu.
Return to Prior Problem.
Continue to Next Lesson.
14
Slopes 1. What is a slope? 2. Slope formula 3.
Types of slopes
15
What is slope? Slope is the slant of a
line. Slope rise change in ys run change in
xs Slope is a fraction/integer.
Lesson Start
16
How to determine the slope when the line goes up.
1. Count the number of units up from the right
point to the left point.
5
4
9
1
2
3
4
5
6
7
8
3
6
2
5
4
1
x
3
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
2
2. Put that number on top of the fraction line.
-1
Slope 6
1
-2
9
-3
-4
3, Count the number of units to the right.
4. Put that number under the fraction line.
y
Lesson Start
17
How to determine the slope when the line goes
down.
1. Count the number of units down from right
point to left point.
5
4
3
-1
-2
2
1
-3
x
-4
2. Put that number on top of fraction line.
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-5
-1
Slope -6
-6
2
1
3
-2
3
-3
-4
3. Count the units to the right.
y
4. Put that number under the fraction line.
Lesson Start
18
Determine the slope of the line shown.
5
4
3
2
1
x
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-1
-2
-3
-4
-1/3
3/1
y
-3/1
1/3
Lesson Start
19
The line does not go down. Go back and try again.
INCORRECT ANSWER
Lesson Start
20
The line does not rise 3 units, then run 1 unit
to the right. Go back and try again.
INCORRECT ANSWER
Lesson Start
21
Determine the slope of the line shown.
5
4
3
2
1
x
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-1
-2
-3
-4
-2/3
3/2
y
-3/2
2/3
Lesson Start
22
The line does not go up. Go back and try again.
INCORRECT ANSWER
Lesson Start
23
The line does not rise -2 units, then run 3 units
to the right. Go back and try again.
INCORRECT ANSWER
Lesson Start
24
Slope Formula m (y1 - y2)
(x1 - x2) where m slope and (x1, y1),
(x2, y2) are points on the line.
Lesson Start
25
Example Find the slope for a line with points
(-3, 4) and (7, -2) on it.
1. Assign values as follows
(x1, y1) (-3, 4) (x2, y2) (7, -2)
2. Substitute them into the formula and solve.
m 4 - (-2) 6 -3
-3 - 7 -10 5
Lesson Start
26
Find the slope of the line with points (5, 6) and
(2, 9) on it. 1. (x1, y1)
(5, 6)
(2, 9)
Lesson Start
27
Find the slope of the line with points (5, 6) and
(2, 9) on it. 1. (x1, y1) (5, 6)
2. (x2, y2)
(5, 6)
(2, 9)
Lesson Start
28
Find the slope of the line with points (5, 6) and
(2, 9) on it. 1. (x1, y1) (5, 6)
2. (x2, y2) (2, 9)
6 9 5 2
6 - 9 5 - 2
3. m
5 - 2 6 - 9
5 2 6 9
Lesson Start
29
The slope formula is a case of subtraction on top
and bottom. Go back and try again.
INCORRECT ANSWER
Lesson Start
30
You have your xs and ys upside down. Remember
Ys guys are always in the skies! Go back and
try again.
INCORRECT ANSWER
Lesson Start
31
You have your xs and ys upside down. You are
also adding when you need to subtract. Go back
and try again.
INCORRECT ANSWER
Lesson Start
32
Find the slope of the line with points (5, 6) and
(2, 9) on it. 1. (x1, y1) (5, 6)
2. (x2, y2) (2, 9)
3. m 6 - 9 -3 -1
5 - 2 3
Lesson Start
33
Find the slope of the line with points (7, 5) and
(3, -4) on it. m
5 - 4 1 7 - 3 4
7 - 3 4 5 - (-4) 9
5 - (-4) 9 3 - 7 -4
5 - (-4) 9 7 - 3 4
Lesson Start
34
You have your xs and ys upside down. Remember
Ys guys are always in the skies! Go back and
try again.
INCORRECT ANSWER
Lesson Start
35
It is not 5 - 4, it is 5 - (-4). Go back and try
again.
INCORRECT ANSWER
Lesson Start
36
You must start with the y and x from the first
point and end with the y and x from the second
point. Go back and try again.
INCORRECT ANSWER
Lesson Start
37
Types of Slopes 1. Positive and Negative
Slopes 2. Special Types of Slopes 3. Determining
Types of Slopes by Looking at Graphs of
Lines 4. Determining Types of Slopes
Algebraically.
Lesson Start
38
Positive and Negative Slopes. Type Graph
Algebra
Positive Up left to right. Positive Frac
tion
SMILE
Negative Down left to right.
Negative Fraction
Frown
Lesson Start
39
2 Special Types of Slopes Type Graphs Algebra
Zero Horizontal Line 0/a, a ? 0
Why, o y, do I look upon the horizon?
Undefined Vertical Line a/0 No Slope
Lesson Start
40
Determining Types of Slopes by Looking at
Graphs of Lines
Lesson Start
41
Is the slope of the line positive, negative,
zero, or undefined?
5
4
3
2
1
x
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-1
-2
-3
-4
y
-
0
?

Lesson Start
42
Is the slope of the line positive, negative,
zero, or undefined?
5
4
3
2
1
x
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-1
-2
-3
-4
y
-
0
?

Lesson Start
43
Is the slope of the line positive, negative,
zero, or undefined?
5
4
3
2
1
x
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-1
-2
-3
-4
y
-
0
?

Lesson Start
44
Is the slope of the line positive, negative,
zero, or undefined?
5
4
3
2
1
x
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-1
-2
-3
-4
y
-
0
?

Lesson Start
45
Click on the line with the negative slope.
5
4
3
2
1
x
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-1
-2
-3
-4
y
Lesson Start
46
Click on the line with the zero slope.
5
4
3
2
1
x
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-1
-2
-3
-4
y
Lesson Start
47
Click on the line with the no slope.
5
4
3
2
1
x
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-1
-2
-3
-4
y
Lesson Start
48
Click on the line with the positive slope.
5
4
3
2
1
x
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-1
-2
-3
-4
y
Lesson Start
49
Determining Types of Slopes Algebraically.
Lesson Start
50
Is the slope of the line with the 2 points listed
positive, negative, zero, or undefined? Points
(-3, 5) and (-9, -4)

-
0
?
Lesson Start
51
Is the slope of the line with the 2 points listed
positive, negative, zero, or undefined? Points
(3, 5) and (3, -4)

-
0
?
Lesson Start
52
Is the slope of the line with the 2 points listed
positive, negative, zero, or undefined? Points
(-3, -5) and (-9, -4)

-
0
?
Lesson Start
53
Is the slope of the line with the 2 points listed
positive, negative, zero, or undefined? Points
(-3, -4) and (-9, -4)

-
0
?
Lesson Start
54
Return to Main Menu.
Return to Prior Problem.
Continue to Next Lesson.
55
Slope-intercept Form of a Linear Equation 1.
Slope-intercept equation 2. Graphing by
slope-intercept 3. Writing slope-intercept
equations
56
Slope-intercept Form y mx b where m
slope and b y-intercept.
Lesson Start
57
Example y -1/2x 4 -1/2 m slope 4
b y-intercept
Lesson Start
58
Graphing by Slope-intercept 1. Graphing lines
with positive/negative slopes. 2. Graphing
lines with zero or undefined/no slopes.
Lesson Start
59
The Slope-intercept Song You make the last number
first. Its either up or down. Make the slope in
2 numbers, Or you look like a clown. Top ones
up or down, And the bottoms always right. Youd
better do it well, Or youll get a fright.
(Tune Hokey-Pokey)
Lesson Start
60
Graph y -1/2x 4
4. Draw a line through the 2 points you plotted.
1. Last number is 4, so go up 4 on the y-axis
from the origin and plot a point.
2. Slope is already 2 numbers. Top one is -1, so
go down 1 from the point you just plotted.
5
4
3
3
2
2
1
1
x
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-1
3. The bottom number is 2, so go 2 units to the
right and plot a point.
-2
-3
-4
y
Lesson Start
61
Graph y 2x - 3
1
4. Go 1 unit to the right and plot a point.
1. Last number is -3, so go down 3 units from the
origin and plot a point.
2. The slope is only 1 number so put a 1 under
the 2.
5
4
3
2
1
x
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
1
-1
3. Go up 2 from the point you just plotted.
5. Draw a line through the 2 points you plotted.
2
-2
-3
-4
y
Lesson Start
62
Click on the graph for y 2/3x - 2
5
4
3
2
1
x
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-1
-2
-3
-4
y
Lesson Start
63
The slope is not negative. Go back and try again.
INCORRECT ANSWER
Lesson Start
64
You graphed the last number on the x-axis instead
of the y-axis. Go back and try again.
INCORRECT ANSWER
Lesson Start
65
Top number is 2, and the bottom is 3, so you do
not go up 3 and over 2. Go back and try again.
INCORRECT ANSWER
Lesson Start
66
Click on the graph for y -4x 3
5
4
3
2
1
x
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-1
-2
-3
-4
y
Lesson Start
67
The slope is not positive. Go back and try again.
INCORRECT ANSWER
Lesson Start
68
The -4 is not the y-intercept, nor is the 3 the
x-intercept. Go back and try again.
INCORRECT ANSWER
Lesson Start
69
The -4 is the slope, not the x-intercept. Go back
and try again.
INCORRECT ANSWER
Lesson Start
70
Two Special Graphs Line with a zero
slope And Line with an undefined slope.
Lesson Start
71
Line with a zero slope y (no
x) graphs as a horizontal line.
Why, o y, do I look upon the horizon?
Lesson Start
72
Graph the equation y 2.
5
4
3
2
1
x
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-1
-2
-3
-4
y
Lesson Start
73
Line with an undefined/no slope x (no
y) graphs as a vertical line.
Lesson Start
74
Graph the equation x -4.
5
4
3
2
1
x
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-1
-2
-3
-4
y
Lesson Start
75
Click on the graph for x 3.
5
4
3
2
1
x
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-1
-2
-3
-4
y
Lesson Start
76
You chose the graph for x -3. Go back and try
again.
INCORRECT ANSWER
Lesson Start
77
The x (no y) line does not graph as a
horizontal line. Go back and try again.
INCORRECT ANSWER
Lesson Start
78
Click on the graph for y -3½.
5
4
3
2
1
x
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-1
-2
-3
-4
y
Lesson Start
79
The y (no x) line does not graph as a
vertical line. Go back and try again.
INCORRECT ANSWER
Lesson Start
80
You chose the graph for y 3½. Go back and try
again.
INCORRECT ANSWER
Lesson Start
81
Writing Slope-intercept Equations 1. When given
a slope and the y-intercept. 2. When given a
slope and one point on the line. 3. When given 2
points on the line.
m ¾, b -1
m -¼, (8, 3)
(3, 7), (5, 12)
Lesson Start
82
Writing a slope-intercept equation when given a
slope and the y-intercept. Substitute the slope
and the y-intercept for the m and b in the
equation. Example m ¾, b -1 y mx b
Slope-int. equation y ¾x - 1 The new
equation
Lesson Start
83
1. Click on the correct equation for a line with
slope 5/3 and y-intercept 2.
y 5/3x 2
y 2x 5/3
5/3y 2x
y -5/3x 2
2. Click on the correct slope and
y-intercept pair for y 7x - 5.
m 7, b -5
m -5, b 7
m -7, b 5
m 1/7, b -5
Lesson Start
84
Writing a slope-intercept equation when given a
slope and a point on the line. 1. Substitute the
x, y, and m in the slope-intercept form. 2.
Solve for b. 3. Substitute the slope and the b in
a clean slope-intercept form.
Lesson Start
85
Example Write the equation of the line
with slope -¼ and point (8, 3).
y mx b 3 -¼(8) b
1. Substitute the slope, x, and y in the equation.
2. Solve for b.
3 -2 b 2 2 5 b
3. Substitute the slope and b in the equation.
y -¼x 5
Lesson Start
86
1. Click on the correct substitution for a line
with slope 1/3 and point (5, 9).
9 1/3(5) b
5 1/3(9) b
9 1/3x 5
9y 5x 1/3
2. Click on the correct equation for a line with
slope -2/3 and point (-6, 4).
y -2/3x
Y -2/3x 4
y -2/3
y -2/3x - 6
Lesson Start
87
Writing a slope-intercept equation for a line
with 2 points given 1. Find the slope of the
line. 2. Use that slope and the first point
to find the y-intercept. 3. Substitute the slope
and the y-intercept into the equation.
Lesson Start
88
Example Write and equation for a line with
points (3, 7) (5, 12).
m (y1 - y2) (x1 - x2) m 7 - 12 -5
5 3 - 5 -2 2
1. Find the slope of the line.
Continued on next screen.
Lesson Start
89
Write and equation for a line with points (3, 7)
(5, 12). m 5/2 y mx b
7 (5/2)(3) b 2(7) 2(15/2) 2b 14
15 2b -15 -15 -1 2b ? -1/2 b 2
2
2. Use the slope and the first point to
solve for the y-intercept.
Continued on next screen.
Lesson Start
90
Write and equation for a line with points (3, 7)
(5, 12). m 5/2, b -1/2
3. Substitute the slope and the y-intercept for
the m and the b in the equation.
y mx b y 5/2x - 1/2
Lesson Start
91
1. Click on the slope for a line with
points (-2, 8) and (7, -5).
13 -9
3 -9
-9 13
13 9
2. Click on the y-intercept for a line
with points (-2, 8) and (7, -5).
46 9
86 9
-5
86 13
Lesson Start
92
3. Click on the correct equation for a line with
points (3, 7) and (4, 8).
y 3x 7
y 3/4x 8
y x 4
y -x 4
Lesson Start
93
Return to Main Menu.
Return to Prior Problem.
Continue to Next Lesson.
94
Graphing by x- and y-intercepts. X-intercept
where the line crosses the x-axis. Y-intercept
where the line crosses the y-axis.
x-intercept
x
y-intercept
y
95
How to graph by x- y-intercepts 1. Cover the x
term with your index finger and solve the
resulting equation for y. 2. Go up or down on the
y-axis from the origin that many units and plot a
point. 3. Cover the y term with your index finger
and solve the resulting equation for x. 4. Go
left or right on the x-axis from the origin that
many units and plot a point. 5. Draw a line
through your points.
Lesson Start
96
Graph the line for 3x 2y 6.
1. Cover the x term and solve for y. 3x 2y 6.
2. Go up 3 units on the y-axis.
5. Draw a line through the points plotted.
5
4
3
2
2
y 3
1
1
x
1
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
3. Cover the y term and solve for x. 3x 2y 6.
-1
4. Go right 2 units on the x-axis.
-2
-3
-4
x 2
y
Lesson Start
97
1. Click on the correct intercepts for 3x - 4y
24.
x-int 8 y-int 6
x-int 8 y-int -6
x-int 6 y-int 8
x-int -6 y-int 8
Lesson Start
98
2. Click on the graph of 3x - 6y 12.
5
4
3
2
1
x
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-1
-2
-3
-4
y
Lesson Start
99
3. Click on the correct equation for the line
shown.
5
-9y - 6x -36
4
4x 6y 36
3
2
1
x
-7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-1
6y 4x 36
-6x - 9y -36
-2
-3
-4
y
Lesson Start
100
Return to Main Menu.
Return to Prior Problem.
Continue to Next Lesson.
101
Graphing Linear Inequalities
Type of Line
Solving Inequalities
Where to Shade
102
How to Determine the Type of Line to Draw
103
Choose the type of line for the inequality given.
1. y gt 3x - 2 a. Solid b. Dotted 2. y gt ¼x -
5 a. Solid b. Dotted
Lesson Start
104
Choose the inequality symbol for the line shown.
lt or gt
lt or gt
Lesson Start
105
Choose the inequality symbol for the line shown.
lt or gt
lt or gt
Lesson Start
106
Where to Shade
For Positive, Negative, Zero Slopes
For Undefined or No Slopes
107
Where to Shade for Positive, Negative, and Zero
Slopes
The inequality must be in y ? mx b format.
? can be gt, gt, lt, or lt.
Lesson Start
108
Lesson Start
109
Graph y gt x - 2.
1. Graph the line y x - 2.
x
2. Since y gt, shade above the line.
y
Lesson Start
110
Graph y lt x - 2.
1. Graph the line y x - 2.
x
2. Since y lt, shade below the line.
y
Lesson Start
111
Do you do anything different when the line is
dotted rather than solid?
Lesson Start
112
Not Really
Lesson Start
113
Graph y gt x - 2.
1. Graph the line y x - 2, but make the line
dotted.
x
2. Since y gt, shade above the line.
y
Lesson Start
114
Graph y lt x - 2.
1. Graph the line y x - 2, but make the line
dotted.
x
2. Since y lt, shade below the line.
y
Lesson Start
115
Graph y gt -½x 3
Type of line
Solid
Dotted
x
y
Lesson Start
116
Graph y gt -½x 3
Type of line
Solid
Dotted
x
Shade ___ the line.
y
Above
Below
Lesson Start
117
Graph y gt -½x 3
Type of line
Solid
Dotted
x
Shade ___ the line.
y
Above
Below
Lesson Start
118
Choose the correct inequality for the graph
shown.
y lt 1/3 x 2
y lt 1/3 x 2
x
y gt 1/3 x 2
y gt 1/3 x 2
y
Lesson Start
119
Where to Shade for Undefined or No Slopes
The inequality must be in x ? (no y) format.
? can be gt, gt, lt, or lt.
Lesson Start
120
Lesson Start
121
Graph x gt -2
1. Draw a dotted vertical line at x -2.
x
2. Shade to the right of the line.
y
Lesson Start
122
Graph x lt -2.
1. Graph the line X -2.
x
2. Shade to the left of the line.
y
Lesson Start
123
Graph x gt 3.
Choose type of line.
Solid
Dotted
x
y
Lesson Start
124
Graph x gt 3.
Choose type of line.
Solid
x
Choose where to shade.
Left
Right
y
Lesson Start
125
Graph x gt 3.
Choose type of line.
Solid
x
Choose where to shade.
Right
y
Lesson Start
126
Solving Inequalities You use the same algebraic
methods as solving equations except when you
multiply/divide both sides by the same negative
number. In that case, you switch the direction of
the inequality symbol.
Lesson Start
127
Solve -3x - 2y lt 12.
-3x - 2y lt 12
3x 3x
-2y lt 3x 12
-2 -2 -2
y lt -3/2 x - 6
gt
Lesson Start
128
Choose the correct inequality. 1. 2x 5y gt -10
y lt 2/5 x 2
y gt 2/5 x 2
y gt -2/5 x - 2
y lt -2/5 x - 2
2. 3x - 2y gt 10
y lt -2/3 x - 5
y gt -2/3 x - 5
y gt 2/3 x - 5
y lt 2/3 x - 5
Lesson Start