5-Minute Check on Activity 3-3

1
5-Minute Check on Activity 3-3

• Examine the following graphs and determine how
  many solutions
• Solutions a) b)
  c)
• Which of the graphs above are consistent?
• Solve the systems of equations
• y 2x 1y 3x 2

One None
Infinite
Only a)
3x 2 y 2x 1 3x 2 2x 1 x 2 1
x 3 and y 3(3) 2 7
Click the mouse button or press the Space Bar to
display the answers.
2
Activity 3 - 4

• How Long Can You Live?

3
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

4
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 lt x lt 9)
• Closed interval end points are included ( )
• Open interval end points are not included ( gt lt
  )

5
English Phases to Math Symbols
Math Symbol English Phrases English Phrases English Phrases
At least No less than Greater than or equal to
gt More than Greater than
lt Fewer than Less than
No more than At most Less than or equal to
Exactly Equals Is
x gt 10 x lt 10 x 10 x 10

1. x is greater than 10
2. x is less than 10
3. x is at least 10
4. x is at most 10

6
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.

7
Solving Inequalities
x 3 3 lt 5 3 ? x lt 8 x 6 6 lt 10
6 ? x lt 4 x lt 10 6 ? x lt
4 7 lt x ? x gt 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!)

8
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.

9
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.

10
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 1980 1985 1990 1995 2000
X, years since 75 0 5 10 15 20 25
W(x)
M(x)
77.01 77.54 78.07 78.60 79.13 79.66
68.94 69.94 70.94 71.94 72.94 73.94
11
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) gt 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
12
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) gt 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
13
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) gt W(x)?
  Graph

14
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) gt W(x)?
  Algebraically

M(x) gt W(x) 0.200x 68.94 gt
0.106x 77.01 Substitute
0.200x gt 0.106x 8.07 -
68.94 0.094x gt 8.07
- 0.106x x
gt 85.852 ? 0.094
15
Algebraic Properties

• Given a lt b thenAddition and
  Subtraction POE keeps the inequality true (a ?
  k lt b ? k)Multiplication or Division by a
  positive number keeps the inequality true (ka lt
  kb, if k gt 0)Multiplication or Division by a
  negative number reverses the inequality (ka gt
  kb, if k lt 0)

16
Algebraic Properties Examples

• If x 2 gt 9
• If x 6 8
• If 6x lt 24
• If ½x 3
• If -y gt 5

then x 2 2 gt 9 2 x gt
11 then x 6 6 8 - 6 x
2 then (6x) / 6 lt (24/6) x lt
4 then 2 ? ½x 2?3 x 6 then
-1?-y gt -1?5 y lt -5
17
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 lt 20
X is greater than 10 and less than or equal to 20 10 lt x 20
X is from 10 to 20 inclusive 10 x 20
18
Compound Inequalities Examples

• Solve the following compound inequalities - 4
  lt 3x 5 11 1 lt 3x 2 lt 4

- 5 - 5 - 5 -9 lt 3x
6 ?3 ?3 ?3 -3 lt x 2
2 2 2 3 lt 3x lt 6
?3 ?3 ?3 1 lt x lt 2
19
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 (?)

20
Interval Notation Examples

• Write the inequalities in interval notation 1 lt
  x lt 2 -9 lt x lt 12 x 3
• Write the interval notations as an inequality
  -2 , 4) (2, 8)
  (5, ?)

(1, 2)
(-9, 12)
(-?, 3
-2 x lt 4
2 lt x lt 8
5 lt x
21
Inequalities and Number Lines

• x gt 4
• x lt 3
• x -1
• x 0
• x 2

(4, ?)
(-?, 3)
-1, ?)
(-?, 0
2, 2
22
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