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Welcome to MM150!
Kirsten Meymaris
Wednesday, May 11th
Unit 4
Plan for the hour


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Variation (4.1)
Linear Inequalities (4.2)
Graphing Linear Equalities (4.3)
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Final Project
Final Project Introduction - What is your chosen profession and how
can one of the math concepts we have studied so far be used in
this profession? Keep in mind that you are not committing to a topic
now; however, you may have decided by the time this Unit is over!
1 - Number Theory (Integers, Natural Numbers, Whole numbers, fractions, improper fractions,
decimals, ...)
2 - Sets, subsets and Venn diagrams
3 - Algebra and basic order of operations
4 - Graphs, linear relationships
5 - The Metric system
6 - Geometry (area, volume, surface area)
7 - Probability
8 - Statistics (sampling and displays)
9 - Statistics (averages, standard deviation)
START NOW MAKING A LIST OF TOPICS THAT YOU THINK YOU USE OR WILL USE IN YOUR
PROFFESSION! Unit 9 comes before you know it!!
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
4.1
Variation
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Variation

Variation is an equation that relates one
variable to one or more other variables.
Direct
Indirect
y  kx
k
y
x
Joint
y  kzx
Combined
kzx
y
l
klz
y
x
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Examples: Direct

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

The amount of interest earned on an investment
and the interest rate.
The time required to fill a pool with a hose and
the volume of water coming from hose.
The volume of a balloon and its radius.
The number of people in line and the time it
takes to reach the front of the line.
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Examples: Direct

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

The amount of interest earned on an investment
and the interest rate. I = kR
The time required to fill a pool with a hose and
the volume of water coming from hose. T = kV
The volume of a balloon and its radius. V = kR
The number of people in line and the time it
takes to reach the front of the line. P = kT
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Examples: Inverse

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The running speed of the person and their race
time.
The number of movies rented from Blockbuster
and the daily rental price.
The time it takes for an ice cube to melt and the
temperature of the water.
The pressure of the water through a hose and
the size of the opening in the hose.
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Examples: Inverse


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
The running speed of the person and their race
time. S = k / T
The number of movies rented from Blockbuster
and the daily rental price. M = k / P
The time it takes for an ice cube to melt and the
temperature of the water. TC = k / TW
The pressure of the water through a hose and
the size of the opening in the hose. P = k / H
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Examples: Joint


The area of a triangle and the base and height
of the triangle.
The area of a square and the length and width
of the square.
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Examples: Joint


The area of a triangle and the base and height
of the triangle. A = kbh
The area of a square and the length and width
of the square. A = klw
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Examples: Combined

The number of movies rented from Blockbuster
varies directly with the advertisement budget
and inversely with the daily rental price.

The electrical resistance of a wire varies directly
as its length and inversely as its crosssectional
area.
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Examples: Combined

The number of movies rented from Blockbuster
varies directly with the advertisement budget
and inversely with the daily rental price.
kA
M
P

The electrical resistance of a wire varies directly
as its length and inversely as its crosssectional
area.
kL
R
A
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
4.2
Linear Inequalities
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Symbols of Inequality

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a < b means that a is less than b.
a  b means that a is less than or equal to b.
a > b means that a is greater than b.
a  b means that a is greater than or equal to b.
Find the solution to an inequality by adding,
subtracting, multiplying or dividing both sides by
the same number or expression.
Change the direction of the inequality symbol
when multiplying or dividing both sides of an
inequality by a negative number.
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example: Graphing on the Number Line

Graph the solution set where x is a real number,
on the number line of
x3
x3
x3
x3
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5
-4
-3
-2
-1
0
1
2
3
4
5
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example: Graphing on the Number Line

Graph the solution set where x is a real number,
on the number line of:
x3
x3
x3
x3
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5
-4
-3
-2
-1
0
1
2
3
4
5
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example: Solve and graph the solution

Solve 3x – 8 < 10 and graph the solution set.
3x – 8 < 10
-5
-4
-3
-2
-1
0
1
2
3
4
5
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example: Solve and graph the solution

Solve 3x – 8 < 10 and graph the solution set.
3 x  8  10
3 x  8  8  10  8
3 x  18
3 x 18

3
3
x6

The solution set is all real numbers less than 6.
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example: Solve and graph the solution

Solve and graph the solution set for:
 15  4 x  3
-5
-4
-3
-2
-1
0
1
2
3
4
5
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example: Solve and graph the solution

Solve and graph the solution set for:
 15  4 x  3
 4 x  3  15
 4 x  3  3  15  3
 4 x  12
 4 x  12

4
4
x3
-5
-4
-3
-2
-1
0
1
2
3
4
5
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Compound Inequality

Graph the solution set of the inequality
4 < x  3
where x is a real number
-5
-4
-3
-2
-1
0
1
2
3
4
5
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Compound Inequality continued

Graph the solution set of the inequality
4 < x  3
where x is a real number
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
4.3
Graphing Linear Equations
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Rectangular Coordinate System
y-axis
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
The horizontal line is
called the x-axis.
The vertical line is
called the y-axis.
The point of
intersection is the
origin.
(dependent)
II
I
x-axis
(Independent)
origin
III
IV
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Plotting Points


Each point in the
xy-plane corresponds
to a unique ordered
pair (a, b).
Plot the point (2, 4).
Move 2 units right
Move 4 units up
4 units
2 units
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Examples
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Graphing Linear Equations
1.
2.
3.
By plotting points
By using the x- and y- Intercepts
By using the slope and y-intercept
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Graphing Equations by
Plotting Points
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
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Solve for y
Determine (at least) 3 points
Plot the points
 Connect
the dots!
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Graphing by Plotting Points

Graph the equation
y = 5x + 2
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Graphing by Plotting Points

Graph the equation
y = 5x + 2
x
0
2/5
1
y
2
0
3
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Graphing Equations by
Plotting Points



Solve for y
Determine (at least) 3 points
Plot the points
 Connect
the dots!
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Graphing Equations
with the Intercepts
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

Determine x- and y-intercepts
Plot the points
Plot a checkpoint
 Connect
the dots!
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Graphing Using Intercepts

Find the x-intercept: y = 0, x = ?
Example:
y = 3x + 6
0 = 3x + 6
(? , 0)

Find the y-intercept: x = 0, y = ?
Example: y = 3x + 6
y = 3(0) + 6
(0, ? )
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Graphing Using Intercepts

The x-intercept is found by letting y = 0 and
solving for x.
Example:
y = 3x + 6
0 = 3x + 6
(2, 0)
6 = 3x
2= x

The y-intercept is found by letting x = 0 and
solving for y.
Example: y = 3x + 6
(0, 6)
y = 3(0) + 6
y=6
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example y = -3x + 6
(2, 0)
(0, 6)
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example y = -3x + 6
(2, 0)
(0, 6)
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Graphing Equations
with the Intercepts



Determine x- and y-intercepts
Plot the points
Plot a checkpoint
 Connect
the dots!
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Graphing Linear Equations
1.
2.
3.
By plotting points
By using the x- and y- Intercepts
By using the slope and y-intercept
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Slope

The ratio of the vertical change to the horizontal
change for any two points on the line.
vertical change
rise
Slope =
=
run
horizontal change
y 2  y1
m
x2  x1
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Types of Slope
y
y
m > 0 positive
m < 0 negative
x
1
x
2
y
y
m is undefined!
m=0
3
x
4
x
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Types of Slope
y
y
m > 0 positive
m < 0 negative
x
1
x
2
y
y
m is undefined!
m=0
3
x
4
x
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Types of Slope
y
y
m > 0 positive
m < 0 negative
x
1
x
2
y
y
m is undefined!
m=0
3
x
4
x
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example:
Finding Slope

Find the slope of the line through the points
(5, 3) and (2, 3).
rise
=
run
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example: Finding Slope

Find the slope of the line through the points
(5, 3) and (2, 3).
y 2  y1
m
x2  x1
3  ( 3)
m
2  5
3  3
m
7
0
m
0
7
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example
(5, -3)
(-2, 3)
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
The Slope-Intercept Form of a Line

Slope-Intercept Form of the Equation of the Line
y = mx + b
m is slope
(0, b) is y-intercept
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Graphing Equations by Using the
Slope and y-Intercept
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Solve the equation for y
Manipulate into slope-intercept form
Determine slope and y-intercept
Plot y-intercept
Plot another point with slope
 Connect
the dots!
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example


Graph 2x  3y = 9.
Write in slope-intercept form.
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example


Graph 2x  3y = 9.
Write in slope-intercept form.
2x  3y  9
3 y  2 x  9
3 y 2 x 9


3
3 3
2
y  x 3
3
The y-intercept is (0,3)
and the slope is 2/3.
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example continued


Plot (0,3)
Using slope
move up 2 units and
to the right 3 units.
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example continued

Plot a point at (0,3)
on the y-axis, then
move up 2 units and
to the right 3 units.
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Special Lines

Graph y = -3

Graph x = -3
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Special Lines

Graph y = -3

Graph x = -3
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Examples – Open Forum!
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Examples – Open Forum!
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Thank You!
Remember to
Ask, Ask, Ask!
[email protected]
AIM: kkmeymaris
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.