Compound Linear Inequalities
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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 TI-83+/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]

Half-Open Interval:  (a, b]  is interpreted as a < x < where a is not included, but b is included.

(1, 5]

Half-Open Interval:  [a, b) is interpreted as a < x < b where a is included, but b is not included.

[1, 5)

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


Non-ending 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 -35 C and 120 C.  The coolant will remain in a liquid state as long as the temperature in Celsius satisfies the inequality
 -35 < C < 120.  Write this inequality in degrees Fahrenheit.
 
Solution:
Remember:

Plan:
-- set up inequality
--
substitute for C
-- solve for (isolate) F

The coolant will remain in a liquid state as long as the temperature in Fahrenheit degrees satisfies the inequality
-31 < F < 248



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"