Inequalities Through Writing, Solving, and Graphing
Lesson
Introduction to
Inequalities
[
OBJECTIVE
]
•
The student will write and solve one-step
inequalities that represent mathematical and
real-world situations and graph the solutions
on number lines
[
MY
SKILLS
]
•
Solving equations
•
Writing expressions
•
Writing equations
[
ESSENTIAL
QUESTIONS
]
1.
How does the solution of the inequality
x
+ 6 < 10 differ from the solution of the
equation
x
+ 6 = 10?
2.
When do you use an open circle when
graphing an inequality? A closed circle?
3.
Explain how to check the solution to an
inequality.
[Warm-Up]
Begin by completing the warm-up for this
lesson.
INTRODUCTION TO INEQUALITIES
SOLVE Problem – Introduction
[
LESSON
]
SOLVE
Jennifer and 3 of her friends are going to a
concert. The price of a ticket includes entrance to
the concert, a CD, and a T-shirt. The total cost of
the tickets is more than $48.00. How could you
represent the cost of one ticket?
[
LESSON
]
SOLVE
S
Study the Problem
Underline the question.
[
LESSON
]
SOLVE
Jennifer and 3 of her friends are going to a
concert. The price of a ticket includes entrance to
the concert, a CD, and a T-shirt. The total cost of
the tickets is more than $48.00. How could you
represent the cost of one ticket?
[
LESSON
]
SOLVE
S
Study the Problem
Underline the question.
This problem is asking me to find
the way to represent the cost of one ticket.
INEQUALITY SYMBOLS
Inequality Symbols
What do you notice about the expressions in Column 1?
The expressions are verbal expressions that compare
values.
Inequality Symbols
What do you notice about Column 2?
Column 2 is a translation of the verbal expression in
Column 1 using symbols and numbers
Inequality Symbols
What are the two number values?
6 and 8
Place a centimeter cube on each of the values on the
number line at the bottom of the page.
Inequality Symbols
Which value is less?
6
We can write the relationship between the two values
using the symbol for less than.
6 < 8
6 < 8
Inequality Symbols
What are the two number values?
4 and 2
Place a centimeter cube on each of the values on the
number line at the bottom of the page.
Inequality Symbols
Which value is greater?
4
We can write the relationship between the two values
using the symbol for greater than.
4 > 2
4 > 2
Inequality Symbols
How is the expression in Problem 3 different from
the expressions in Problems 1 and 2?
The expression in Problem 3 contains a variable.
Inequality Symbols
Explain why we use a variable.
We use a variable to represent an unknown value
What are the two values in Problem 3?
x
and 5
Inequality Symbols
What is the relationship between the two?
x
is less than or equal to 5.
We can write that relationship using the sign for less
than or equal to.
x
≤ 5
x
≤ 5
Inequality Symbols
How is the expression in Problem 4 different from
the expressions in Problems 1 and 2?
The expression in Problem 4 contains a variable.
Inequality Symbols
Explain why we use a variable.
We use a variable to represent an unknown value
What are the two values in Problem 4?
c
and 1
Inequality Symbols
What is the relationship between the two?
c
is greater than or equal to 1.
We can write that relationship using the sign for
greater than or equal to.
c
≥ 1
c
≥ 1
INEQUALITY SYMBOLS WITH REAL-
WORLD SITUATIONS
Inequality Symbols with Real-World
Situations
Identify the values in Problem 1.
15 video games and a variable to represent how many
games John has
Let’s use the variable “
j
” to represent the number of
video games that John has.
Inequality Symbols with Real-World
Situations
How can we write the relationship using a variable,
numbers, and symbols?
j
< 15
j
< 15
Inequality Symbols with Real-World
Situations
This relationship is called an inequality.
The two values are not equal. One is less than the
other.
j
< 15
Why is this relationship called an inequality?
Inequality Symbols with Real-World
Situations
Identify the values in Problem 2.
6 friends and a variable to represent how many friends
were invited in total.
Let’s use the variable “
p
” to represent the number of
friends at the birthday party.
Inequality Symbols with Real-World
Situations
How can we write the relationship using a variable,
numbers, and symbols?
p
> 6
p
> 6
Inequality Symbols with Real-World
Situations
Identify the values in Problem 3.
20 students and a variable to represent the total
number of students in the class.
Let’s use the variable “
m
” to represent the number of
students in the math class.
Inequality Symbols with Real-World
Situations
How can we write the relationship using a variable,
numbers, and symbols?
m
≥ 20
m
≥ 20
Inequality Symbols with Real-World
Situations
Identify the values in Problem 4.
$12 and a variable to represent the amount he can
spend at the carnival.
Let’s use the variable “
k
” to represent the amount that
Kyle can spend at the carnival.
Inequality Symbols with Real-World
Situations
How can we write the relationship using a variable,
numbers, and symbols?
k
≤ 12
k
≤ 12
Inequality Symbols with Real-World
Situations
Take a look back at Problem 1.
It is less than 15.
j
< 15
What do we know about the value of
j
?
Inequality Symbols with Real-World
Situations
What do you notice about the numbering on the graph
in Column 3?
It has 5 values and the middle value is 15.
j
< 15
Inequality Symbols with Real-World
Situations
When we graph an inequality, we draw a circle at the
point of the value given in the inequality.
15
j
< 15
Where do we draw the circle?
Draw the circle.
Inequality Symbols with Real-World
Situations
Are we including 15 in our possible answers?
No
j
< 15
When the value is not included, we use an empty circle.
Inequality Symbols with Real-World
Situations
When we have an equation, how many solutions are
there?
1
j
< 15
When we have an inequality, how many solutions are
there?
many
Inequality Symbols with Real-World
Situations
We can show the solution set for the inequality by
drawing a ray from the point on the number line.
j
< 15
Which direction should we draw the ray? Explain your
answer.
We draw the ray from the point at 15 to the left
because our solution must be less than 15.
Inequality Symbols with Real-World
Situations
When we graph an inequality, we draw a circle at the
point of the value given in the inequality.
6
Where do we draw the circle?
Draw the circle.
p
> 6
Inequality Symbols with Real-World
Situations
Are we including 6 in our possible answers?
No
When the value is not included, we use an empty circle.
p
> 6
Inequality Symbols with Real-World
Situations
Which direction should we draw the ray? Explain your
answer.
We draw the ray from the point at 6 to the right
because our solution must be greater than 6.
p
> 6
Inequality Symbols with Real-World
Situations
When we graph an inequality, we draw a circle at the
point of the value given in the inequality.
20
Where do we draw the circle?
Draw the circle.
m
≥ 20
Inequality Symbols with Real-World
Situations
Are we including 20 in our possible answers?
Yes
When the value is included, we use a filled in circle.
m
≥ 20
Inequality Symbols with Real-World
Situations
Which direction should we draw the ray? Explain your
answer.
We draw the ray from the point at 20 to the right
because our solution must be greater than
or equal to 20.
m
≥ 20
Inequality Symbols with Real-World
Situations
When we graph an inequality, we draw a circle at the
point of the value given in the inequality.
12
Where do we draw the circle?
Draw the circle.
k
≤ 12
Inequality Symbols with Real-World
Situations
Are we including 12 in our possible answers?
Yes
When the value is included, we use a filled in circle.
k
≤ 12
Inequality Symbols with Real-World
Situations
Which direction should we draw the ray? Explain your
answer.
We draw the ray from the point at 12 to the left because
our solution must be less than
or equal to 12.
k
≤ 12
Inequality Symbols with Real-World
Situations
How are Problems 5 – 8 different from Problems 1 – 4?
The inequality is given, but no situation.
x
< 3
What are some possible real-world scenarios for
Problem 5?
Sample: there are less than 3 cookies for each student.
SAMPLE: There are
less than 3 cookies
for each student
Inequality Symbols with Real-World
Situations
How can we graph this inequality? Where will we place
the circle?
Above the 3
x
< 3
Is the circle open or filled in? Explain your answer.
The circle will be open because the inequality tells us
the solution set is less than 3 and does not include the
value of 3.
SAMPLE: There are
less than 3 cookies
for each student
Inequality Symbols with Real-World
Situations
Which direction does the ray extend?
To the left because the solution set is values less than 3
x
< 3
Draw the ray.
SAMPLE: There are
less than 3 cookies
for each student
ONE-STEP INEQUALITIES –
ADDITION AND SUBTRACTION
One-Step Inequalities – Addition and
Subtraction
Begin by solving the addition and subtraction
equations for Problems 1 and 2.
One-Step Inequalities – Addition and
Subtraction
– 2 – 2
x
= 2
2 + 2 = 4
4 = 4
One-Step Inequalities – Addition and
Subtraction
+ 2 + 2
x
= 6
6 − 2 = 4
4 = 4
One-Step Inequalities – Addition and
Subtraction
How did you find the solution for the equation in
Problem 1?
Isolate the variable by subtracting 2 from both sides.
What was the solution for this equation?
The answer for this equation was
x
= 2.
x
+ 2 = 4
One-Step Inequalities – Addition and
Subtraction
Look at the problem in the second column. How is this
problem different from the problem in Column 1?
x
+ 2 < 4
It has an inequality symbol instead of an equals sign.
Inequalities can be solved using the same process as
the one used to solve equations.
One-Step Inequalities – Addition and
Subtraction
What two things need to happen each time that we
solve an equation?
x
+ 2 < 4
•
Isolate the variable
•
Balance the equation
One-Step Inequalities – Addition and
Subtraction
In solving inequalities we will also need to isolate the
variable and whatever we do to one side of the
inequality we must do to the other.
x
+ 2 < 4
One-Step Inequalities – Addition and
Subtraction
– 2 – 2
x
= 2
2 + 2 = 4
4 = 4
We can isolate the variable by subtracting 2.
How can we isolate the variable in the first inequality?
One-Step Inequalities – Addition and
Subtraction
– 2 – 2
x
= 2
2 + 2 = 4
4 = 4
Because this is an addition inequality, we use the
inverse operation of subtraction to solve it.
Explain your answer.
One-Step Inequalities – Addition and
Subtraction
– 2 – 2
x
= 2
2 + 2 = 4
4 = 4
What will you need to do to the other side of the
inequality?
Also subtract 2
– 2 – 2
x
< 2
One-Step Inequalities – Addition and
Subtraction
– 2 – 2
x
= 2
2 + 2 = 4
4 = 4
What is the value of
x
in the inequality?
x
< 2
– 2 – 2
x
< 2
One-Step Inequalities – Addition and
Subtraction
– 2 – 2
x
= 2
2 + 2 = 4
4 = 4
We can check the answer for an inequality using the
same process as the one we used to check the answer
for an equation.
– 2 – 2
x
< 2
One-Step Inequalities – Addition and
Subtraction
– 2 – 2
x
= 2
2 + 2 = 4
4 = 4
Take a look back at the equation from Problem 1.
– 2 – 2
x
< 2
One-Step Inequalities – Addition and
Subtraction
– 2 – 2
x
= 2
2 + 2 = 4
4 = 4
How many values were there for the variable
x
?
1
– 2 – 2
x
< 2
One-Step Inequalities – Addition and
Subtraction
– 2 – 2
x
= 2
2 + 2 = 4
4 = 4
In an inequality, there are often several values that will
make the statement true. Look at the inequality above.
– 2 – 2
x
< 2
One-Step Inequalities – Addition and
Subtraction
– 2 – 2
x
= 2
2 + 2 = 4
4 = 4
What was the solution for the inequality in Problem 1?
x
< 2
– 2 – 2
x
< 2
One-Step Inequalities – Addition and
Subtraction
– 2 – 2
x
= 2
2 + 2 = 4
4 = 4
What does this mean?
Any value that is less than 2 should work when
substituted back into the original inequality.
– 2 – 2
x
< 2
One-Step Inequalities – Addition and
Subtraction
– 2 – 2
x
= 2
2 + 2 = 4
4 = 4
Let’s substitute the value of 1 into the original inequality.
The value 1, which is less than 2, makes this a true
statement because 3 is less than 4.
– 2 – 2
x
< 2
1 + 2 < 4
3 < 4 True
One-Step Inequalities – Addition and
Subtraction
Choose a value that is greater than 2, such as 5, to try
in the original inequality. Any value greater than 2 will
make the inequality not true.
x
+ 2 < 4
5 + 2 < 4
7 < 4
Not a true statement because 7 is not
less than 4
ONE-STEP INEQUALITIES –
MULTIPLICATION AND DIVISION
One-Step Inequalities – Multiplication
and Division
Begin by solving the multiplication and division
equations for Problems 3 and 4.
One-Step Inequalities – Multiplication
and Division
2 2
x
= 2
2(2) = 4
4 = 4
One-Step Inequalities – Multiplication
and Division
x
= 4
One-Step Inequalities – Multiplication
and Division
How did we find the solution for the this equation?
We had to isolate the variable by dividing both sides
by 2.
2 2
x
= 2
2(2) = 4
4 = 4
One-Step Inequalities – Multiplication
and Division
Look at the problem in the Inequality column. How is
this problem different from the problem in Column 1?
2
x
< 4
It has an inequality symbol instead of an equals sign.
Inequalities can be solved using the same process as
with equations.
One-Step Inequalities – Multiplication
and Division
What two things need to happen each time we solve
an equation?
•
Isolate the variable
•
Balance the equation
2
x
< 4
In solving inequalities we will also need to isolate the
variable and whatever we do to one side of the
inequality we must do to the other.
One-Step Inequalities – Multiplication
and Division
2 2
x
= 2
2(2) = 4
4 = 4
We can isolate the variable by dividing by 2.
How can we isolate the variable in the first inequality?
One-Step Inequalities – Multiplication
and Division
What will we need to do to the other side of the
inequality?
Also divide by 2
2 2
x
= 2
2(2) = 4
4 = 4
2 2
x
< 2
One-Step Inequalities – Multiplication
and Division
What is the value of
x
in the inequality?
x
< 2
2 2
x
= 2
2(2) = 4
4 = 4
2 2
x
< 2
One-Step Inequalities – Multiplication
and Division
We can check the answer for an inequality using the
same process as the one we used to check the answer
for an equation.
2 2
x
= 2
2(2) = 4
4 = 4
2 2
x
< 2
One-Step Inequalities – Multiplication
and Division
Look back at Problem 3.
2 2
x
= 2
2(2) = 4
4 = 4
2 2
x
< 2
One-Step Inequalities – Multiplication
and Division
How many values were there for the variable
x
?
1
2 2
x
= 2
2(2) = 4
4 = 4
2 2
x
< 2
One-Step Inequalities – Multiplication
and Division
In an inequality, there are often several values that will
make the statement true.
2 2
x
= 2
2(2) = 4
4 = 4
2 2
x
< 2
One-Step Inequalities – Multiplication
and Division
What is the solution for the inequality in Problem 3?
x
< 2
2 2
x
= 2
2(2) = 4
4 = 4
2 2
x
< 2
One-Step Inequalities – Multiplication
and Division
Therefore, any value that is less than 2 should work
when substituted back into the original inequality.
2 2
x
= 2
2(2) = 4
4 = 4
2 2
x
< 2
One-Step Inequalities – Multiplication
and Division
Substitute the value of 1 back into the original
inequality. The value of 1, which is less than 2, makes
this a true statement because 2 is less than 4.
2 2
x
= 2
2(2) = 4
4 = 4
2 2
x
< 2
2(1) < 4
2 < 4 True
One-Step Inequalities – Multiplication
and Division
Choose a value that is greater than 2, such as 5, to try
in the original inequality. Any value greater than 2 will
make the inequality not true.
2
x
< 4
2(5) < 4
10 < 4
Not a true statement because 10 is not
less than 4
GRAPHING INEQUALITIES
Graphing Inequalities
– 2 – 2
x
= 2
2 + 2 = 4
4 = 4
What is the solution for the inequality in Problem 1?
x
< 2
– 2 – 2
x
< 2
2 + 1 < 4
3 < 4 True
x
< 2
Graphing Inequalities
– 2 – 2
x
= 2
2 + 2 = 4
4 = 4
Because there is more than one value that can be a
solution to the inequality, the solution can be graphed
using a number line.
– 2 – 2
x
< 2
2 + 1 < 4
3 < 4 True
x
< 2
Graphing Inequalities
We can use the solution of the inequality (
x
< 2) to
determine how to number the number line and how to
graph the solution.
SOLUTION:
x
< 2
The solution contains a positive 2, so we can place a 2
in the middle of the number line and label the values
to the left and right of the 2.
Graphing Inequalities
1
0
2
3
4
– 2 – 2
x
= 2
2 + 2 = 4
4 = 4
– 2 – 2
x
< 2
2 + 1 < 4
3 < 4 True
x
< 2
Graphing Inequalities
Is the value of 2 a solution for the inequality? Explain
your thinking.
SOLUTION:
x
< 2
No, because
x
is less than 2.
Begin graphing the inequality by drawing a circle
above the 2. The circle above the 2 is open because 2
is not included in the solution.
Graphing Inequalities
– 2 – 2
x
= 2
2 + 2 = 4
4 = 4
– 2 – 2
x
< 2
2 + 1 < 4
3 < 4 True
x
< 2
Graphing Inequalities
Which direction should the arrow point?
Justify your answer.
SOLUTION:
x
< 2
The solution set includes all the values less than 2, so
the arrow should point to the left.
The arrowhead at the end of the line shows that the
values for the solution will continue to infinity.
Graphing Inequalities
– 2 – 2
x
= 2
2 + 2 = 4
4 = 4
– 2 – 2
x
< 2
2 + 1 < 4
3 < 4 True
x
< 2
INTRODUCTION TO INEQUALITIES
SOLVE Problem – Completion
[
LESSON
]
SOLVE
Jennifer and 3 of her friends are going to a
concert. The price of a ticket includes entrance to
the concert, a CD, and a T-shirt. The total cost of
the tickets is more than $48.00. How could you
represent the cost of one ticket?
[
LESSON
]
SOLVE
S
Study the Problem
Underline the question.
This problem is asking me to find
the way to represent the cost of one ticket.
O
Organize the Facts
Identify the facts.
[
LESSON
]
SOLVE
Jennifer and 3 of her friends are going to a
concert. The price of a ticket includes entrance to
the concert, a CD, and a T-shirt. The total cost of
the tickets is more than $48.00. How could you
represent the cost of one ticket?
O
Organize the Facts
Identify the facts.
Eliminate the unnecessary facts.
[
LESSON
]
SOLVE
Jennifer and 3 of her friends are going to a
concert. The price of a ticket includes entrance to
the concert, a CD, and a T-shirt. The total cost of
the tickets is more than $48.00. How could you
represent the cost of one ticket?
O
Organize the Facts
Identify the facts.
Eliminate the unnecessary facts.
List the necessary facts.
•
Jennifer and 3 friends
•
Total cost
is more than $48.00
L
Line Up a Plan
Write in words what your plan of action will
be.
Write and solve an inequality that I can use to
represent the cost of one ticket.
Choose an operation or operations.
Division
V
Verify Your Plan with Action
Estimate your answer.
About $10.00
Carry out your plan.
4
x
> $48.00
4 4
x
> $12.00
E
Examine Your Results
Does your answer make sense?
(Compare your answer to question.)
Yes, because we were looking for how to
represent the cost of one ticket.
Is your answer reasonable?
(Compare your answer to the estimate.)
Yes, because it is close to my estimate of
about $10.00.
Is your answer accurate?
(Check your work.)
Yes
Write your answer in a complete sentence.
The cost of 1 ticket can be represented by the
inequality
x
> $12.00.
INTRODUCTION TO INEQUALITIES
Closure
[
ESSENTIAL
QUESTIONS
]
1.
How does the solution of the inequality
x
+ 6 < 10 differ from the solution of the
equation
x
+ 6 = 10?
An equation has only one solution, and
an inequality has many solutions.
[
ESSENTIAL
QUESTIONS
]
2.
When do you use an open circle when
graphing an inequality? Closed circle?
Use an open circle when the value is
not a solution to the inequality and a
closed circle when the value is a
solution to the inequality.
[
ESSENTIAL
QUESTIONS
]
3.
Explain how to check the solution to an
inequality.
After solving the inequality, choose any
value that is within the solution set and
substitute it into the original inequality.
Inequality
Inverse Operation(s)
Isolate the Variable
Less Than
Greater Than
Less Than Or Equal To
Greater Than Or Equal To
Inequality Symbols (<, >, ≤, ≥)
Solution
Number Line
Lesson
Introduction to
Inequalities
Delve into the world of inequalities with this lesson, where you will learn to write and solve one-step inequalities for both mathematical and real-world scenarios. By the end, you will be able to graph the solutions on number lines, enhancing your understanding of this fundamental concept in mathematics.
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Presentation Transcript
Lesson Introduction to Inequalities
[OBJECTIVE] The student will write and solve one-step inequalities that represent mathematical and real-world situations and graph the solutions on number lines
[MYSKILLS] Solving equations Writing expressions Writing equations
[ESSENTIALQUESTIONS] 1. How does the solution of the inequality x + 6 < 10 differ from the solution of the equation x + 6 = 10? 2. When do you use an open circle when graphing an inequality? A closed circle? 3. Explain how to check the solution to an inequality.
[Warm-Up] Begin by completing the warm-up for this lesson.
SOLVE Problem Introduction INTRODUCTION TO INEQUALITIES
[LESSON] SOLVE Jennifer and 3 of her friends are going to a concert. The price of a ticket includes entrance to the concert, a CD, and a T-shirt. The total cost of the tickets is more than $48.00. How could you represent the cost of one ticket?
[LESSON] SOLVE S Study the Problem Underline the question.
[LESSON] SOLVE Jennifer and 3 of her friends are going to a concert. The price of a ticket includes entrance to the concert, a CD, and a T-shirt. The total cost of the tickets is more than $48.00. How could you represent the cost of one ticket?
[LESSON] SOLVE S Study the Problem Underline the question. This problem is asking me to find the way to represent the cost of one ticket.
Inequality Symbols Expression Writing with Symbols 1. six is less than eight 2. four is greater than two 3. x is less than or equal to five 4. c is greater than or equal to one 5. three is less than seven 6. twelve is greater than nine 7. b is less than or equal to ten 8. y is greater than or equal to eleven What do you notice about the expressions in Column 1? The expressions are verbal expressions that compare values.
Inequality Symbols Expression Writing with Symbols 1. six is less than eight 2. four is greater than two 3. x is less than or equal to five 4. c is greater than or equal to one 5. three is less than seven 6. twelve is greater than nine 7. b is less than or equal to ten 8. y is greater than or equal to eleven What do you notice about Column 2? Column 2 is a translation of the verbal expression in Column 1 using symbols and numbers
Inequality Symbols Expression 1. six is less than eight Writing with Symbols 11 12 13 14 15 10 6 9 5 7 8 4 3 2 1 What are the two number values? 6 and 8 Place a centimeter cube on each of the values on the number line at the bottom of the page.
Inequality Symbols Expression 1. six is less than eight Writing with Symbols 6 < 8 11 12 13 14 15 10 6 9 5 7 8 4 3 2 1 Which value is less? 6 We can write the relationship between the two values using the symbol for less than. 6 < 8
Inequality Symbols Expression 2. four is greater than two Writing with Symbols 11 12 13 14 15 10 6 9 5 7 8 4 3 2 1 What are the two number values? 4 and 2 Place a centimeter cube on each of the values on the number line at the bottom of the page.
Inequality Symbols Expression 2. four is greater than two Writing with Symbols 4 > 2 11 12 13 14 15 10 6 9 5 7 8 4 3 2 1 Which value is greater? 4 We can write the relationship between the two values using the symbol for greater than. 4 > 2
Inequality Symbols Expression 3. x is less than or equal to five Writing with Symbols 11 12 13 14 15 10 6 9 5 7 8 4 3 2 1 How is the expression in Problem 3 different from the expressions in Problems 1 and 2? The expression in Problem 3 contains a variable.
Inequality Symbols Expression 3. x is less than or equal to five Writing with Symbols 11 12 13 14 15 10 6 9 5 7 8 4 3 2 1 Explain why we use a variable. We use a variable to represent an unknown value What are the two values in Problem 3? x and 5
Inequality Symbols Expression 3. x is less than or equal to five Writing with Symbols x 5 11 12 13 14 15 10 6 9 5 7 8 4 3 2 1 What is the relationship between the two? x is less than or equal to 5. We can write that relationship using the sign for less than or equal to. x 5
Inequality Symbols Expression 4. c is greater than or equal to one Writing with Symbols 11 12 13 14 15 10 6 9 5 7 8 4 3 2 1 How is the expression in Problem 4 different from the expressions in Problems 1 and 2? The expression in Problem 4 contains a variable.
Inequality Symbols Expression 4. c is greater than or equal to one Writing with Symbols 11 12 13 14 15 10 6 9 5 7 8 4 3 2 1 Explain why we use a variable. We use a variable to represent an unknown value What are the two values in Problem 4? c and 1
Inequality Symbols Expression 4. c is greater than or equal to one Writing with Symbols c 1 11 12 13 14 15 10 6 9 5 7 8 4 3 2 1 What is the relationship between the two? c is greater than or equal to 1. We can write that relationship using the sign for greater than or equal to. c 1
INEQUALITY SYMBOLS WITH REAL- WORLD SITUATIONS
Inequality Symbols with Real-World Situations Expression Inequality Graph 1. John has less than 15 video games. 15 16 17 14 13 Identify the values in Problem 1. 15 video games and a variable to represent how many games John has Let s use the variable j to represent the number of video games that John has.
Inequality Symbols with Real-World Situations Expression Inequality Graph 1. John has less than 15 video games. j < 15 15 16 17 14 13 How can we write the relationship using a variable, numbers, and symbols? j < 15
Inequality Symbols with Real-World Situations Expression Inequality Graph 1. John has less than 15 video games. j < 15 15 16 17 14 13 This relationship is called an inequality. Why is this relationship called an inequality? The two values are not equal. One is less than the other.
Inequality Symbols with Real-World Situations Expression Inequality Graph 2. Penny invited more than 6 friends to her birthday party. 8 5 6 7 4 Identify the values in Problem 2. 6 friends and a variable to represent how many friends were invited in total. Let s use the variable p to represent the number of friends at the birthday party.
Inequality Symbols with Real-World Situations Expression Inequality Graph 2. Penny invited more than 6 friends to her birthday party. p > 6 8 5 6 7 4 How can we write the relationship using a variable, numbers, and symbols? p > 6
Inequality Symbols with Real-World Situations Expression Inequality Graph 3. There are at least 20 students in the math class 20 21 22 19 18 Identify the values in Problem 3. 20 students and a variable to represent the total number of students in the class. Let s use the variable m to represent the number of students in the math class.
Inequality Symbols with Real-World Situations Expression Inequality Graph 3. There are at least 20 students in the math class m 20 20 21 22 19 18 How can we write the relationship using a variable, numbers, and symbols? m 20
Inequality Symbols with Real-World Situations Expression Inequality Graph 4. Kyle could spend no more than $12 at the school carnival 12 13 14 11 10 Identify the values in Problem 4. $12 and a variable to represent the amount he can spend at the carnival. Let s use the variable k to represent the amount that Kyle can spend at the carnival.
Inequality Symbols with Real-World Situations Expression Inequality Graph 4. Kyle could spend no more than $12 at the school carnival k 12 12 13 14 11 10 How can we write the relationship using a variable, numbers, and symbols? k 12
Inequality Symbols with Real-World Situations Expression Inequality Graph 1. John has less than 15 video games. j < 15 15 16 17 14 13 Take a look back at Problem 1. What do we know about the value of j? It is less than 15.
Inequality Symbols with Real-World Situations Expression Inequality Graph 1. John has less than 15 video games. j < 15 15 16 17 14 13 What do you notice about the numbering on the graph in Column 3? It has 5 values and the middle value is 15.
Inequality Symbols with Real-World Situations Expression Inequality Graph 1. John has less than 15 video games. j < 15 15 16 17 14 13 When we graph an inequality, we draw a circle at the point of the value given in the inequality. Where do we draw the circle? 15 Draw the circle.
Inequality Symbols with Real-World Situations Expression Inequality Graph 1. John has less than 15 video games. j < 15 15 16 17 14 13 Are we including 15 in our possible answers? No When the value is not included, we use an empty circle.
Inequality Symbols with Real-World Situations Expression Inequality Graph 1. John has less than 15 video games. j < 15 15 16 17 14 13 When we have an equation, how many solutions are there? 1 When we have an inequality, how many solutions are there? many
Inequality Symbols with Real-World Situations Expression Inequality Graph 1. John has less than 15 video games. j < 15 15 16 17 14 13 We can show the solution set for the inequality by drawing a ray from the point on the number line. Which direction should we draw the ray? Explain your answer. We draw the ray from the point at 15 to the left because our solution must be less than 15.
Inequality Symbols with Real-World Situations Expression Inequality Graph 2. Penny invited more than 6 friends to her birthday party. p > 6 8 5 6 7 4 When we graph an inequality, we draw a circle at the point of the value given in the inequality. Where do we draw the circle? 6 Draw the circle.
Inequality Symbols with Real-World Situations Expression Inequality Graph 2. Penny invited more than 6 friends to her birthday party. p > 6 8 5 6 7 4 Are we including 6 in our possible answers? No When the value is not included, we use an empty circle.
Inequality Symbols with Real-World Situations Expression Inequality Graph 2. Penny invited more than 6 friends to her birthday party. p > 6 8 5 6 7 4 Which direction should we draw the ray? Explain your answer. We draw the ray from the point at 6 to the right because our solution must be greater than 6.
Inequality Symbols with Real-World Situations Expression Inequality Graph 3. There are at least 20 students in the math class m 20 20 21 22 19 18 When we graph an inequality, we draw a circle at the point of the value given in the inequality. Where do we draw the circle? 20 Draw the circle.
Inequality Symbols with Real-World Situations Expression Inequality Graph 3. There are at least 20 students in the math class m 20 20 21 22 19 18 Are we including 20 in our possible answers? Yes When the value is included, we use a filled in circle.
Inequality Symbols with Real-World Situations Expression Inequality Graph 3. There are at least 20 students in the math class m 20 20 21 22 19 18 Which direction should we draw the ray? Explain your answer. We draw the ray from the point at 20 to the right because our solution must be greater than or equal to 20.
Inequality Symbols with Real-World Situations Expression Inequality Graph 4. Kyle could spend no more than $12 at the school carnival k 12 12 13 14 11 10 When we graph an inequality, we draw a circle at the point of the value given in the inequality. Where do we draw the circle? 12 Draw the circle.
Inequality Symbols with Real-World Situations Expression Inequality Graph 4. Kyle could spend no more than $12 at the school carnival k 12 12 13 14 11 10 Are we including 12 in our possible answers? Yes When the value is included, we use a filled in circle.
Inequality Symbols with Real-World Situations Expression Inequality Graph 4. Kyle could spend no more than $12 at the school carnival k 12 12 13 14 11 10 Which direction should we draw the ray? Explain your answer. We draw the ray from the point at 12 to the left because our solution must be less than or equal to 12.
Inequality Symbols with Real-World Situations Expression Inequality Graph 5. SAMPLE: There are less than 3 cookies for each student x < 3 5 2 3 4 1 How are Problems 5 8 different from Problems 1 4? The inequality is given, but no situation. What are some possible real-world scenarios for Problem 5? Sample: there are less than 3 cookies for each student.
Inequality Symbols with Real-World Situations Expression Inequality Graph 5. SAMPLE: There are less than 3 cookies for each student x < 3 5 2 3 4 1 How can we graph this inequality? Where will we place the circle? Above the 3 Is the circle open or filled in? Explain your answer. The circle will be open because the inequality tells us the solution set is less than 3 and does not include the value of 3.
