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5-Minute Check on Activity 3-3
Solve the systems of equations:
1. 2y – 2 = 4x
y = 3x – 2
by substitution:
2(3x – 2) – 2 = 4x
6x – 4 – 2 = 4x
6x – 6 = 4x
-6 = -2x
3=x
and y = 3(3) – 2 = 7
2. 6y – 2x = 8
4y + 2x = 2
by addition:
6y – 2x = 8
4y + 2x = 2
10y
= 10
y=1
and
4(1) + 2x = 2
4 + 2x = 2
2x = -2
x = -1
Click the mouse button or press the Space Bar to display the answers.
Activity 3 - 4
How Long Can You Live?
Objectives
• Solve linear inequalities in one variable
numerically and graphically
• Use properties of inequalities to solve linear
inequalities in one variable algebraically
• Solve compound inequalities in one variable
algebraically and graphically
• Use interval notation to represent a set of
real numbers by an inequality
Vocabulary
• Inequality – a relationship in which one side can be
greater or less than the other (equal as well)
• Compound inequality – an inequality involving to
inequality signs (like 3 < x < 9)
• Closed interval – end points are included (≥ ≤)
• Open interval – end points are not included ( > < )
English Phases to Math Symbols
Math
Symbol
≥
>
<
≤
=
English Phrases
At least
No less than Greater than or equal to
More than
Greater than
Fewer than
Less than
No more than At most
Less than or equal to
Exactly
Equals
Is
1. x is greater than 10
x > 10
2. x is less than 10
x < 10
3. x is at least 10
x ≥ 10
4. x is at most 10
x ≤ 10
Solving Inequalities
Solving an inequality in one variable is the process of
determining the values of the variable that make the
inequality a true statement. These values are called
the solutions of the inequality.
If we had the solution to a pair of equalities (lines from
previous lessons), then it was the point of intersection.
With inequalities, if we have a solution, then we have a
region of lots of points that satisfy the inequalities.
We will use the same properties of Equality to solve the
inequalities algebraically.
Solving Inequalities
x–3+3<5+3

x<8
x + 6 – 6 < 10 – 6

x<4
x < 10 – 6

x<4
7<x

x>7
• Any action you apply to one side of an inequality
must be applied to the other side to keep the
inequality in balance
–
–
–
–
We can add the same number to both sides
We can subtract the same number from both sides
We can simplify one or both sides
We cannot interchange the sides (we flip the inequality!)
How Long Can You Live?
Life expectancy in the United States is steadily
increasing, and the number of Americans aged 100 or
older will exceed 850,000 by the middle of this century.
Medical advancements have been a primary reason for
Americans living longer. Another factor has been the
increased awareness of maintaining a healthy lifestyle.
How Long Can You Live?
The life expectancies at birth for women and men born
after 1975 can be modeled by the following functions:
W(x) = 0.106x + 77.01
M(x) = 0.200x + 68.94
where W(x) represents the life expectancy for women,
M(x) represents the life expectancy for men, and x
represents the number of years since 1975 that the
person was born. That is, x = 0 corresponds to the year
1975, x = 5 corresponds to 1980, and so forth.
How Long Can You Live? (cont)
The life expectancies at birth for women and men born
after 1975 can be modeled by the following functions:
W(x) = 0.106x + 77.01
M(x) = 0.200x + 68.94
Complete the following table:
1975
X, years since 75
W(x)
M(x)
0
1980
5
1985
1990
1995
2000
10
15
20
25
77.01 77.54 78.07 78.60 79.13 79.66
68.94 69.94 70.94 71.94 72.94 73.94
How Long Can You Live? (cont)
The life expectancies at birth for women and men born
after 1975 can be modeled by the following functions:
W(x) = 0.106x + 77.01
M(x) = 0.200x + 68.94
When will men overtake women in life expectancies?
When will M(x) > W(x)?
We can solve it one of three ways:
1) Using a table in our calculator
2) Using the graphing capability of our calculator
3) Solve it algebraically
How Long Can You Live? - Table
The life expectancies at birth for women and men born
after 1975 can be modeled by the following functions:
W(x) = 0.106x + 77.01
M(x) = 0.200x + 68.94
When will men overtake women in life expectancies?
When will M(x) > W(x)?
Table
X
Y1
Y2
83
85.808
85.54
84
85.914
85.74
85
86.02
85.94
86
86.126
86.14
87
86.232
86.34
How Long Can You Live? - Graph
The life expectancies at birth for women and men born
after 1975 can be modeled by the following functions:
W(x) = 0.106x + 77.01
M(x) = 0.200x + 68.94
When will men overtake women in life expectancies?
When will M(x) > W(x)?
Graph
y
x
10 20 30 40 50 60 70 80 90
How Long Can You Live? - Alg
The life expectancies at birth for women and men born
after 1975 can be modeled by the following functions:
W(x) = 0.106x + 77.01
M(x) = 0.200x + 68.94
When will men overtake women in life expectancies?
When will M(x) > W(x)?
Algebraically
M(x) > W(x)
0.200x + 68.94 > 0.106x + 77.01
0.200x > 0.106x + 8.07
0.094x > 8.07
x > 85.852
Substitute
- 68.94
- 0.106x
 0.094
Algebraic Properties
• Given
a<b
then
Addition and Subtraction POE keeps the
inequality true (a  k < b  k)
Multiplication or Division by a positive number
keeps the inequality true (ka < kb, if k > 0)
Multiplication or Division by a negative number
reverses the inequality (ka > kb, if k < 0)
Algebraic Properties: Examples
• If x – 2 > 9
then x – 2 + 2 > 9 + 2
x > 11
• If x + 6 ≤ 8
then x + 6 – 6 ≤ 8 - 6
x≤2
• If 6x < 24
then (6x) / 6 < (24/6)
x<4
• If ½x ≥ 3
then 2  ½x ≥ 23
x≥6
• If -y > 5
then -1-y > -15
y < -5
Compound Inequalities
• When a variable is between two numbers,
then it is called a compound inequality
• Remember the English translations!
• Examine the following table:
Statement in English
Compound Inequality
X is greater than or equal to 10,
but less than 20
10 ≤ x < 20
X is greater than 10 and less than
or equal to 20
10 < x ≤ 20
X is from 10 to 20 inclusive
10 ≤ x ≤ 20
Compound Inequalities: Examples
• Solve the following compound inequalities:
- 4 < 3x + 5 ≤ 11
-5
-5 -5
-9 < 3x
3
-3 <
≤ 6
3
3
x
≤ 2
1 < 3x – 2 < 4
+2
+2 +2
3 < 3x
< 6
3
3
3
1 <
x
< 2
Interval Notation
• Closed Interval: denoted by [ ]
means the endpoints are included
• Open Interval: denoted by ( )
means the endpoints are not included
• Half open or Half closed: denoted by ( ] or [ )
means one endpoint is included and the
other is not (base on open and closed above)
• Unbounded: denoted by -  or 
means that an interval can go as low as
negative infinity (- ) or that an interval can
go as high as positive infinity ()
Interval Notation: Examples
• Write the inequalities in interval notation:
1<x<2
-9 < x < 12
x≤3
(1, 2)
(-9, 12)
(-, 3]
• Write the interval notations as an inequality:
[-2 , 4)
(2, 8)
(5, )
-2 ≤ x < 4
2<x<8
5<x
Inequalities and Number Lines
• x>4
(4, )
• x<3
(-, 3)
• x ≥ -1
-1, )
• x≤0
(-, 0]
• x=2
[2, 2]
1
2
3
4
5
6
7
1
2
3
4
5
6
7
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
1
2
3
4
5
6
7
Summary and Homework
• Summary
– The solution set of an inequality in one variable is
the set of all values of the variable that satisfy the
inequality.
– The direction of an inequality is not changed
when
• Same quantity is added to or subtracted from
both sides of the inequality, or
• Both sides of an inequality are multiplied or
divided by the same positive number.
• Homework
– pg 330 – 335; 4-8, 19, 20