Transcript Venn Diagram - Bibb County Schools

Algebra 1 Unit 2:
Learning Goals 2.1, 2.2 & 2.3
Graphing and Analyzing One
Variable Linear Inequalities
Michelle A. O’Malley
League Academy of Communication Arts
Greenville, South Carolina
Learning Goal 2.1
 Write and solve one, two, and multistep inequalities in one variable, and
graph the solution set on the number
line, including real-world applications.
Learning Goal 2.1 Standards
 EA1.3: Apply algebraic methods to solve
problems in real-world contexts
 EA4.8: Carry out procedures to solve linear
inequalities for one variable algebraically
and then to graph the solution
 EA5.12 Analyze given information to write a
linear inequality in one variable that models
a given problem situation.
Learning Goal 2.1 Word Wall Words
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Inequality
Greater than
Less than
Equal to
Not equal to
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Solution set
Infinite
Discrete
Continuous
Inequality Review
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>
≥
<
≤
≠
means
means
means
means
means
greater than
greater than or equal to
less than
less than or equal to
is not equal to
 Proper way to read the above information
would be 3 < X means “x is greater than 3”
Solution Set
 A solution set of a linear inequality
in one variable is the set of all values
that satisfy the inequality. (The
statement is true)
 Note: an inequality has an infinite
number of solutions.
(Infinite means never ends)
Discrete Vs. Continuous
 A domain may be discrete or continuous
 Discrete data is data that can be described by
whole numbers or fractional values. Example:
the data has an ending point such as whole
numbers, which are not continuous or
repeating.
 Continuous data is data that relate to a
complete range of values on the number line.
Example: The possible sizes of applies are
continuous data
Discrete Vs. Continuous
 The graph of a solution can be continuous
on a number line (a ray). For example, at
what temperatures will the ice in a skating
pond remain frozen?
 The graph of a solution can also be discrete
set of points on a number line. For
example, a club must have 25 or more
members.
Graphing Inequalities
 > means greater than
 < means less than
 When the above two inequalities are graphed, the graph
will show an open dot at the endpoint.
 This means that the value at that position is not included
in the solution set.
 ≥ means greater than or equal to
 ≤ means less than or equal to
 When the above two inequalities are graphed, the graph
will show a solid dot at the endpoint.
 This means that the value at that position is included in
the solution set.
Solving Inequalities
 When solving inequalities you follow the
same methods that you used as when
solving equations; however, there is one
exception.
 When you multiply or divide both sides of the
inequality by the same negative number, the
direction of the inequality reverses.
Solving Inequalities
 This is because multiplying a number by -1
moves it to the other side of the zero on
the number line ( a reflection).
 For example
 When multiplying 2 < 5 (a true statement) by
-1, it will force the inequality to change to -2 > -5
to keep the statement true.
Solving Inequalities
 However, a negative sign in an inequality
does not necessarily mean that the
direction of the inequality symbol should
change. For example, when solving
x/6 > -3, do not change the direction of
the inequality. Can you see why?
Real World Applications Example 1
 Nick’s car averages 18 miles per
gallon of gasoline. If x represents the
number of gallons, how much
gasoline will he need to travel at least
450 miles?
 18x ≥ 450
 X ≥ 25
 He will have to buy 25 or more gallons.
Real World Applications Example 2
 An amusement park charges $5 for
admission and $1.25 for each ride. Katie
goes to the park with $25. If x represents
the number of rides, what is the maximum
number of rides that she can afford?
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5 + 1.25x ≤ 25
1.25x ≤ 20
X ≤ 16
Katie can ride at most 16 rides.
Solving Inequalities by
Addition and Subtraction
Solving Inequalities by
Multiplication and Division
Solving Multi-Step
Inequalities
Learning Goal 2.2
 Use Venn Diagrams to represent
unions and intersections of sets.
Learning Goal 2.2: Standards
 EA1.6: Understand how algebraic
relationships can be represented to
concrete models, pictorial models,
and diagrams
Learning Goal 2.2 Word Wall Words
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Universal Set
Intersection
Union
Disjoint
Intersecting
Venn Diagram
Learning Goal 2.2 Word Wall Words
 Intersection – the set of elements that
belong to each of two overlapping sets.
 Union – a set that is formed by combining
the members of two or more sets, as
represented by the symbol U. The Union
contains all members previously contained
in either set.
 Disjoint – Not connected
 Venn Diagram – a pictorial means of
representing the relationships between
sets.
Venn Diagrams
 An element belongs to the intersection of two
sets A and B (written A  B) if and only if the
element belongs to set A and to set B.
 An element belongs to the union of two sets A
and B (written A U B) if and only if the element
belongs to set A or to set B or both A and B.
Venn Diagram
 A Venn Diagram shows relationships
among sets of data or objects. It usually
consists of a rectangle that represents
the universal set. Circles are show
inside the rectangle to represent subsets
of the universal set.
Venn Diagram
 Set can be related to each other in three
ways as illustrated in the diagrams
below.
 The sets can be separate (disjoint),
overlapping (intersecting), or within each
other (subsets).
A
U
A
B
U
B
Venn Diagram
 Sets A and B are disjoint since they
have no elements in common.
 For example, the universal set (U)
represents the set of all polygons.
 Circle A represents the set of all
parallelograms.
 Circle B represents the set of all
trapezoids
U
A
B
r2
r3
r1
Venn Diagram
 Sets A and B intersect. For example, the
universal set (U) represents the set of
polygons.
 Circle A represents the set of all rectangles.
 Circle B represents the set of all rhombuses.
 The intersection of the two circles is the set of
all squares, which represents polygons that are
both rectangles and rhombuses.
A
r2
U
r4
r1
B
r3
Venn Diagram
 The set B is a subset of A since all
elements in B are also elements of A.
 For example, the universal set (U)
represents the set of all polygons.
 Circle A represents the set of all
rectangles.
 Circle B represents the set of all squares.
A
r2
U
B
r3
r1
Venn Diagram
 In the figure below, Region A  B is
represented by
(the empty set).
 A U B is represented by all the elements in
region 2 combined with all the elements in
the region 3.
U
A
B
r2
r3
r1
Venn Diagram
 In the figure below, region A  B is
represented by elements in region 4.
 A U B is represented by all the
elements in region 2 combined with all
the elements in regions 3 and 4.
A
r2
U
r4
r1
B
r3
Venn Diagram
In the figure below, region A  B
is represented by region 3. A U B
is represented by all the elements
in region 2, which includes the
elements in region 3.
A
r2
U
B
r3
r1
Venn Diagram
 In the figure below, region 6 (r6)
represents the intersection of all three sets.
 For example, if the three circles represent
people who like chocolate, vanilla, and
strawberry ice cream, region 6 represents
those people who like all three flavors.
V
r8
r7
r4
S
r6
r3
r5
r2
C
r1
Using Venn Diagrams to
Solve Problems
Learning Goal 2.3
 Write and solve compound
inequalities and graph the solution
set on the number line including realworld applications.
Learning Goal 2.3: Standards
 EA-1.5 Demonstrate an
understanding of algebraic
relationships by using a variety of
representations (including verbal,
graphic, numerical, and symbolic).
 EA-4.8 Carry out procedures to solve
linear inequalities for one variable
algebraically and then to graph the
solution. (Extension)
Learning Goal 2.3 Word Wall Words
 Compound Inequality
 Intersection
 Union
Learning Goal 2.3
 A compound inequality is two or
more inequalities joined with the
world “and” or with the word “or.”
 To solve a compound inequality with
the word “and” (a conjunction),
solve each simple inequality, then
find the solution that make both
simple inequalities true.
 This solution is the intersection of
the solution sets of the simple
inequalities.
Learning Goal 2.3
 3 < X < 5 is equivalent to 3 < x and x
< 5, then it is also true that 3 < 5.
 It is considered good form to write the
smallest numbers on the left as shown
above.
 Compound conjunctions can be solve
by either working with the combined
form or by separating it into two
simple inequalities and finding the
intersection of the two solutions.
Learning Goal 2.3
 To solve a compound inequality
with the word “or” (a
disjunction), solve each simple
inequality, then find the solutions
that make either simple inequality
true.
 The solution is the union of the
solution sets of the simple
inequalities.
Learning Goal 2.3
 “AND” means Intersection and also
conjunction
 This means that the solution must be found in both
inequalities; therefore the solution is the common
area of your graphs.
 “OR” means union and Disjunction
 This means that the solution will be found in one or
the other inequalities; therefore the solution will be
ALL areas graphed combined for both inequalities.
Learning Goal 2.3
Solve and Graph a Union
Step 1: Solve
each individual
inequality
Step 2: Graph
each individual
inequality
solution
Step 3: for a
Union your
compound
inequality
solution will be
all sections
combined
Learning Goal 2.3
Solve and Graph an Intersection
Step 1: solve each
individual
inequality
Step 2: Graph
each individual
inequality
Step 3: The
solution for the
intersection will
be the area
that each
graph has in
common
Work Cited
 Carter, John A., et. al. Glencoe
Mathematics Algebra I. New
York: Glencoe/McGraw-Hill, 2003.
 Greenville County Schools Math
Curriculum Guide
 Gizmos
http://www.explorelearning.com
 Math Use’s Handbook: Hot Words Hot
Topics. New York: Glencoe McGrawHill, 1 998.