
Remember the
primary rule for working with linear inequalities:
... multiplying or dividing by a negative number
changes the direction of the inequality. 

Let's extend our knowledge of inequalities to more sophisticated problems and
more applied situations.

You will now be seeing more references
to interval notation when working with linear inequalities. Check
the table below if you need a quick review of this notation. 

How to use your
TI83+/84+ graphing calculator with one variable inequalities.
Click calculator. 


Interval Notation:
(description) 
(diagram) 
Open
Interval: (a, b) is
interpreted as a < x < b where the endpoints
are NOT included.
(While this notation resembles an ordered
pair, in this context it refers to the interval upon
which you are working.) 
(1, 5)

Closed
Interval: [a, b] is interpreted
as a < x < b where the endpoints
are included. 
[1, 5]

HalfOpen
Interval: (a, b] is interpreted
as a < x < b where a is not included,
but b is included. 
(1, 5]

HalfOpen
Interval: [a, b) is interpreted as
a
< x < b where a is included, but
b is not
included. 
[1, 5)

Nonending
Interval:
is interpreted as x > a where
a is
not included and infinity is always expressed as
being "open" (not included). 

Nonending
Interval:
is interpreted as x < b where
b is included and again, infinity is always
expressed as being "open" (not included). 


A
compound
inequality is two simple inequalities joined by "and" or
"or".
Solving an
"And" Compound Inequality: 
3x  9 < 12 and 3x
 9 > 3 
Also written ...

Or written ...

The common statement
is sandwiched
between the two inequalities.
Solve as a single unit or solve each side
separately. 
The solution is 2 < x
< 7,
which can be read
x >
2 and x
< 7.
Interval notation:
[2, 7] 



Solving an
"Or" Compound Inequality: 
2x + 3 < 7 or
5x + 5 > 25 
Also written ...
[2x + 3 < 7]
[5x + 5 > 25] 

Solve the first inequality 

Solve the
second inequality 
The solution is x < 2
or
x > 4.
Interval notation: 


Applied Problem
Using "AND" 
The antifreeze added to your car's cooling system claims
that it will protect your car to 35º C and 120º C.
The coolant will remain in a liquid state as long as the
temperature in Celsius satisfies the inequality
35º < C < 120º. Write this inequality in degrees Fahrenheit.

Applied Problem
Using "AND" and "OR" 
The height of a horse is
measured in a vertical line from the ground to the withers
(at the base of the neck). Horses are measured in
"hands" where one hand = 4 inches. If a horse is more
than an exact number of hands high (hh), the extra inches
are given after a decimal point, e.g. 14 hands 2 inches is
written as 14.2 hh. Normal riding horses are between
14.3 hh and 17 hh, inclusive. Horses shorter than 14.3
hands are called ponies and horses over 17 hh are often
called draft (or work) horses.
a.)
Write an inequality statement to represent the
heights of normal riding
horses in inches.
b.) Write an inequality statement
stating the heights, in inches, of equines
(horses) that do not fit the normal riding
horse height specifications. 

Solution:
a.) Normal riding
horse heights in hands:
14.3 hh
< h < 17 hh 
Convert to inches.
14.3 hh = 14(4) + 3 inches
= 59
inches
17 hh = 17(4) inches
= 68 inches 
Answer:
Normal riding horse height in inches:
59" < h < 68" 

b.)
Equines outside of the normal riding
horse heights in hands:
h < 14.3 hh or h > 17
hh 
Use conversions from part a. 
Answer: Equine
heights in inches not fitting the normal riding
horse heights:
h < 59" or
h > 68" 


