Graph Systems of Linear
Inequalities
Peter Paliwoda
Multi-step video game problem
• An online media store is having a sale, as
described in the ad shown. Use the information
in the ad to write and graph a system of
inequalities for the regular video game prices and
possible sale prices. Then use the graph to
estimate the range of possible sale prices for
games that are regularly priced at $20.
ONE DAY SALE!
SAVE 30%-70%
ON ALL
VIDEO GAMES
REGULAR PRICE: $20-$50)
System of Inequalities
•
•
•
•
x>=20
x<=50
y>=0.3x
y<=0.7x
Regular price must be at least $20
Regular price can be at most $50
Sale price is at least (100-70)%=30% of regular price
Sale price is at most (100-30)%=70% of regular price
ONE DAY SALE!
SAVE 30%-70%
ON ALL
VIDEO GAMES
REGULAR PRICE: $20-$50)
Graph Inequalities
Sale Price (dollars)
y
x>=20
x<=50
x
Regular Price (dollars)
Identify the Range of Possible Sale Prices
• From the graph you can see that when x=$40,
the value of y is between these values:
0.3(40)=12 and 0.7(40)=28
So, the value of y satisfies 12 <= y <= 28
Modeling Bicycle
• Speed of a Bicycle: The radii of the pedal
sprocket, the wheel sprocket, and the wheel
of the bicycle in the figure are 4 inches, 2
inches, and 14 inches respectively. A cyclist is
pedaling at a rate of 1 revolution per second.
Find Speed of Bicycle in ft/min
pedal _ sprocket
2    r  2    4 "  25 . 13 " / sec
wheel _ sprocket
2    r  2    2 "  12 . 57 " / sec
pedal _ sprocket

wheel _ sprocket
25 . 13
2
12 . 57
wheel _ of _ bicycle
2    r  2    14 "  88 "
2 _ wheel _ turns _ for _ each _ pedal _ sprocket _ turn
speed _ of _ bicycle 
( 2    14  2 )"
sec

176 " 1 ft 14 . 7 ft


sec 12 "
sec
Write function for Distance d (in miles)
in terms of n revolutions of the pedal
n  2   4
2   2
2
 2    14  ( n  2  2    14 )" 
1mile
63360 "
 n  ( 0 . 002776 ) miles
Write a function for the distance d (in
miles) a cyclist travels in terms of time
(in seconds).
n  2   4
2   2
 2    14  ( n  2  2    14 )" 
1mile
63360 "

n
 ( 0 . 002776 ) miles
sec
 n  ( 0 . 002776 )
miles
sec
Why should you learn Inverse
functions
• It is useful in manipulating graphs and the way
they are displayed
• Recall that to find the inverse of a line of
points f(x)={(1,5),(2,6),(3,7),(4,8)} is done by
interchanging the first and second coordinate
• f-1(x)={(5,1),(6,2),(7,3),(8,4)}
• So, f is equal to the range of f-1 and vice versa.
Problem
• The factory sales f (in millions of dollars) of
digital cameras in the United States from 1998
to 2003 are shown in the table.
Year, t
Sales, f(t) (in millions)
1998
519
1999
1209
2000
1825
2001
1972
2002
2794
2003
3421
Questions
Year, t
Sales, f(t) (in millions)
1998
519
1999
1209
2000
1825
2001
1972
2002
2794
2003
3421
• Does f-1 exist?
• If f-1 exists, what does it represent in the context
of the problem?
• If f-1 exists, find f-1 (1825)
• If the table was extended to year 2004 and if the
factory sales of digital cameras for that year was
$2794 million, would f-1 exist? Explain
Does f-1 exist? Yes, it does exist!
f(x)
Year, t
Sales, f(t) (in millions)
1998
519
1999
1209
2000
1825
2001
1972
2002
2794
2003
3421
f-1(x)
Sales, f(t) (in millions)
Year, t
519
1998
1209
1999
1825
2000
1972
2001
2794
2002
3421
2003
If f-1 exists, what does it represent in the context
of the problem?
• f-1 represents the domain of f
in f(x)
x
f(x)
Domain
Range
in f-1(x)
x
f-1(x)
Domain of f-1(x)
Range of f-1(x)
Range of f(x)
Domain of f(x)
If f-1 exists, find f-1 (1825)
Year, t
Sales, f(t) (in millions)
1998
519
1999
1209
2000
1825
2001
1972
2002
2794
2003
3421
f-1(x)
Sales, f(t) (in millions)
Year, t
519
1998
1209
1999
1825
2000
1972
2001
2794
2002
3421
2003
If the table was extended to year 2004 and if
the factory sales of digital cameras for that year
was $2794 million, would f-1 exist? Explain
• No, because the extended function would fails
the horizontal line test. More than one
horizontal intersection