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Lessons 21, Section 4.1
Inequalities
Ex 1: Which of the following numbers would be solutions of the inequality shown?
4 x  5  2( x  1)
0, -5, -3, 12
4(0)  5  2(0  1)
0  5  2(1)
true
5  2
4(5)  5  2(5  1)
 20  5  2(6)
false
 15  12
4(3)  5  2(3  1)
 12  5  2(4)
true
 7  8
4(12)  5  2(12  1)
48  5  2(11)
true
53  22
Solutions include 0, -3, and 12.
There are three ways to represent an inequality; using the inequality symbol (set-builder
notation), a number-line graph, and using interval notation. The examples below
represent equivalent forms of all three ways.
inequality symbol
number-line graph
interval notation
{x | x  5}
(,5)
5
{x | x  2}
[2, )
2
{x | 0  x  3}
(0, 3)
0
3
{x | 4  x  10}
[4, 10]
4
10
{x | 3  x  1}
(3,1]
-3
1
1
There is a summary of these ways on the course webpage (other information,
inequalities).
*Do not confuse interval notation with an ordered pair (point). The context in which
each is used will make the meanings clear.
{ y | y  2}
a)
Write in interval notation and graph on the number line.
b)
Write using set-builder notation and using interval notation.
-5
c)
Write using set-builder notation and graph on the number line.
[0,5)
Begin with the following inequality: 20 > 12
Do the following operations to both
sides of the inequality and determine if the result is true or false.
add 4
subtract 3
multiply by 4
divide by 2
multiply by -2
divide by -4
20 + 4 > 12 + 4
20 - 3 > 12 - 3
4(20) > 4(12)
20 12

2
2
-2(20) > -2(12)
20 12

4 4
true
true
true
true
?
?
Solving Inequalities: When solving an inequality you may add, subtract, multiply
by a positive number, or divide by a positive number on both sides and the result is
true. However, if you multiply or divide by a negative number, the inequality sign
must be reversed!
Solve these inequalities. Write the answer using both set-builder notation and interval
notation, then graph the solution.
Ex 2:
x  12  5
Note: In place of an open circle, we now use a
parenthesis. In place of a closed circle, we now
use a brachet. Both are to open in the direction
of the shading.
2
4
10
x
5
11
Ex 3:

Ex 4:
3a  1
 7
2
Ex 5: 6(2 y  8)  3( y  10)
Ex 6: 7(b  2)  6b  3(3  6b)
3
Ex 7: 13  (2c  2)  2(c  2)  3c
Ex 8:
1
1
 (6 x  24)  20  (12 x  72)
3
4
Ex 9:
2
3
(5 x  1)  (4 x  2)  2
3
4
4
Applications of Inequalities
Phrases commonly translated to inequalities:
Words
is at least
Sample Sentence
Gina is at least 18 years old
Translation
g  18
are at most
There are at most 12 coins
in his pocket.
The total value cannot
exceed $12.
The car's speed must exceed
20 miles per hour
There can be no more than
50 people in the room.
The nut mix can have no
less than 5 cups of almonds.
Marie's age is less than 21
c  12
cannot exceed
must exceed
no more than
no less than
is less than
is more than
is between
The house is more than
35500 square feet.
The distance is between 50
and 200 miles.
v  12
s  20
p  50
n5
m  21
A  35500
50  d  200
Examples:
1)
Metro Concerts can rent a truck for either $55 with unlimited mileage or $29 plus
40¢ per mile. For what mileages would the unlimited mileage plan save money?
Let m = numbers of miles
5
2)
A long-distance phone call using East Calling costs 10 cents for the first minute
and 8 cents for each additional minute. The same call on West Call System cost
15 cents for the first minute and 6 cents for each additional minute. For what
length phone calls is East Calling less expensive than West Call System?
Let m = number of total minutes
3)
Musclebound Movers charges $85 plus $40 an hour to move households across
town. Champion Movers charges $60 an hour for cross-town moves. For what
lengths of time is Champion more expensive?
Let h = number of hours for the move
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