# Introduction to inequalities

```Teaching & Learning Plans
Inequalities
Junior Certificate Syllabus
The Teaching & Learning Plans
are structured as follows:
Aims outline what the lesson, or series of lessons, hopes to achieve.
Prior Knowledge points to relevant knowledge students may already have
and also to knowledge which may be necessary in order to support them in
accessing this new topic.
Learning Outcomes outline what a student will be able to do, know and
understand having completed the topic.
Relationship to Syllabus refers to the relevant section of either the Junior and/
or Leaving Certificate Syllabus.
Resources Required lists the resources which will be needed in the teaching
and learning of a particular topic.
Introducing the topic (in some plans only) outlines an approach to introducing
the topic.
Lesson Interaction is set out under four sub-headings:
i.
Student Learning Tasks – Teacher Input: This section focuses on possible lines
of inquiry and gives details of the key student tasks and teacher questions which
move the lesson forward.
ii.
Student Activities – Possible Responses: Gives details of possible student
reactions and responses and possible misconceptions students may have.
iii. Teacher’s Support and Actions: Gives details of teacher actions designed to
support and scaffold student learning.
iv. Assessing the Learning: Suggests questions a teacher might ask to evaluate
whether the goals/learning outcomes are being/have been achieved. This
evaluation will inform and direct the teaching and learning activities of the next
class(es).
Student Activities linked to the lesson(s) are provided at the end of each plan.
2
Teaching & Learning Plans:
Inequalities
Aims
The aim of this series of lessons is to enable students to:
• enable students to understand the relationship between numbers
• enable students to represent inequalities on the number line
• enable students to solve linear inequalities and relate these to everyday life
Prior Knowledge
Students have prior knowledge of:
• Sets
• Number systems
• How to represent all number systems on the number line
• Order of numbers on the number line
• Patterns including: completing tables and drawing graphs of patterns
• Linear equations in one unknown
Learning Outcomes
As a result of studying this topic, students will be able to:
• determine if a number is less than, less than or equal to, greater than or
greater than or equal to another number
• represent solutions to inequalities on the number line
• simplify and solve linear inequalities by table, graph and/or formula
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Teaching & Learning Plan: Inequalities
Catering for Learner Diversity
In class, the needs of all students, whatever their level of ability level, are
equally important. In daily classroom teaching, teachers can cater for different
abilities by providing students with different activities and assignments graded
according to levels of difficulty so that students can work on exercises that
match their progress in learning. Less able students, may engage with the
activities in a relatively straightforward way while the more able students
should engage in more open-ended and challenging activities.
In interacting with the whole class, teachers can make adjustments to meet
the needs of all of the students. For example, some students may engage with
some of the more challenging questions for example question 6 in Section H:
Student Activity 1.
Apart from whole-class teaching, teachers can utilise pair and group work to
encourage peer interaction and to facilitate discussion. The use of different
grouping arrangements in these lessons should help ensure that the needs
of all students are met and that students are encouraged to articulate their
mathematics openly and to share their learning.
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Teaching & Learning Plan: Inequalities
Relationship to Junior Certificate Syllabus
Topic
Description of topic
Learning outcomes
Students should be able to
4.7 Equations Using a variety of
Solve linear inequalities in
and
problem solving strategies one variable of the form
Inequalities
to solve equations
g(x) ≤ k where
and inequalities. They
identify the necessary
g(x) = ax + b,
information, represent
problems mathematically, a ∈ N and b, k, ∈ Z;
making correct use of
symbols, words, diagrams,
tables and graphs.
k ≤ g(x) ≤ h where
g(x) = ax + b and
k, a, b, h, ∈ Z and
x∈R
Resources Required
Graph matching exercises and Tarsias contained in this Teaching and Learning
Plan need to be laminated and cut up.
For Graph matching exercises see Section C: Student Activity 1 page 37 and
Section I: Student Activity 3 page 58.
For Tarsias see Section D: Student Activity 1 page 41 and Section I: Student
Activity 2 page 56.
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Teaching & Learning Plan: Inequalities
Lesson Interaction
Student Activities: Possible Teacher’s Supports and Checking Understanding
and Expected Responses
Actions
Teacher Reflections
Section A: Revision of <, >, ≤ and ≥ symbols.
»» Which would you prefer to have, 4
euro or 6 euro?
• 6
»» Is 4 less than or greater than 6?
• Less than
»» Is 3 less than or greater than 1?
• Greater than
»» Can you think of words or phrases that
mean less than?
• Smaller
»» Can you think of words or phrases that
mean greater than?
• Bigger
»» In mathematics we use the “<” symbol
to represent less than.
»» Write “4 is less than
6” on the board.
»» Do students
understand the
difference between
»» Write “3 is greater
less than and greater
than 1” on the board.
than?
»» Write the following
table on the board.
»» Look at the shape of the symbol. The
shape of this symbol goes from small
to big.
Less than
<
Greater than
>
»» We use the “>” symbol to represent
greater than. The shape of this symbol
goes from big to small.
»» Write 8 is greater than 5 in symbol
format.
• 8 > 5
»» Write 4 is less than 5 in symbol format.
• 4 < 5
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»» Do students know that
< means less than and
> means greater than?
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Teaching & Learning Plan: Inequalities
Student Activities: Possible
and Expected Responses
Teacher’s Supports and
Actions
Checking
Understanding
»» Is -3 less than or greater than 4?
• -3 < 4
»» Which would you prefer to owe your
friend, €3 or €2?
• €2
»» Write each expression on
the board in words and
symbols.
»» Do students
understand that
-3 < -2 etc.?
»» Is -3 less than or greater than -2?
• -3 < -2
Is -3 less than or greater than -4?
• -3 > -4
»» Is -9 less than or greater than -7?
• -9 < -7
»» Is 5 < or > 5?
• Neither, 5 is equal to 5
»» Do questions 1 and 2 in Section A:
Student Activity 1.
• Students complete
questions 1 and 2 in Section
A: Student Activity 1.
»» Mathematicians use ≤ to represent
less than or equal to and ≥ to
represent greater than or equal to.
These are also known as the inclusive
inequalities.
»» Can you think of other words or
phrases that might be used to describe
less than or equal to?
• At most
No more than
»» Can you think of other words or
phrases that might be used to describe
greater than or equal to?
• At least
Not less than
»» Complete the table on the board.
• Students complete the table
on the board.
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Teacher Reflections
»» Distribute Section A:
Student Activity 1.
»» Draw the table containing
the following words on
symbols when students
to complete the table in
their exercise books.
Less than
<
Greater than
>
Less than or equal to
≤
Greater than or equal to
≥
KEY: » next step
»» Do students
understand
the difference
between less
than and less
than and equal
to?
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Teaching & Learning Plan: Inequalities
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Supports and
Actions
Checking Understanding
»» When the statement uses
the symbols <, >, ≤ or ≥ we
call it an inequality and
when = is used we call it an
equation.
»» Do students understand
what is meant by an
inequality and how it
differs from an equation?
»» Can students use the ≤ and
≥ symbols correctly?
»» Write, using the symbols
above, examples of
inequalities and equations.
• Students write out their
own examples.
»» Ask students to come to the
board to present their own
examples.
»» How many solutions does
the equation x = 7 have?
• One, since the equation
states that x is equal to 7.
»» Circulate the room and offer »» Can students convert
assistance to students when
context based questions to
required.
mathematical language?
»» How many solutions does
the inequality x > 7, x ∈ N
have?
»» Emily has at least €200 in
her savings account. How
would you write this using
mathematical symbols?
• Infinite, x could be 8, 9,
10, …
•
s ≥ 200
Teacher Reflections
»» Show students where to
find questions 3 and 4 in
Section A: Student Activity
distributed.
»» In pairs, complete questions • In pairs students discuss
3 and 4 in Section A:
and compare their
Student Activity 1.
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Teaching & Learning Plan: Inequalities
Teacher Input
Student Activities: Possible Teacher’s Supports and Actions
and Expected Responses
Checking
Understanding
Section B: Revision of Number Systems.
»» Give me some examples of
Natural Numbers?
• {1, 2, 3, 4, 5…}
»» What letter is normally used
to denote the set of Natural
Numbers?
•
»» Can Natural Numbers be
represented on the number
line?
• Yes
»» Give me some examples of
Integers?
• {…-3.-2,-1,0,1,2,3…}
»» What letter is normally used
to denote the set of Integers?
•
»» Are all Natural Numbers
Integers? Give examples.
• Yes. {1, 2, 3…}
»» Can Integers be represented
on the number line?
• Yes
»» Draw a number line and
represent the Natural
Numbers and Integers
represented on the diagram
on the board on this number
line.
• Students draw a number
line and represent the
Natural Numbers and
Integers on it.
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N
»» Draw the following diagram
on the board and add numbers
students suggest where
appropriate.
Teacher Reflections
»» Can students
use sets and
set notation to
show what they
understand by
Natural Numbers?
Z
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»» Can students
represent Natural
Numbers and
Integers on the
number line?
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Teaching & Learning Plan: Inequalities
Student Activities: Possible and
Expected Responses
»» What are Rational Numbers?
• Numbers that can be written as
fractions.
»» Can you give me an example of a
Rational Number?
• For example -¾, 2.25, ½.
»» What letter is normally used to
represent the set of Rational Numbers?
•
»» Is 3 a Rational Number?
• Yes. It can be written as a fraction
i.e. 3/1 or 6/2 etc.
»» Are all Integers Rational Numbers?
Why?
• Yes. All Integers can be written as
fractions e.g. -8 = -24/3 or -16/2.
Also 5 = 10/2 or 20/4 etc.
»» Can Rational Numbers be represented
on the number line?
• Yes
»» Is 2.5 a Rational Number and why?
• Yes. It can be written as a fraction.
»» What are the numbers that cannot be
written as a fraction called?
• Irrational
»» Give examples of Irrational Numbers?
• √2,√3 or π are examples of
irrational numbers. The square root
of every prime number is irrational.
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Teacher’s Supports
and Actions
Checking
Understanding
Teacher Reflections
Q
»» Write 3/1 or 6/2 on the »» Can students
board.
recall how
to write
»» Write the following
equivalent
on the board:
fractions?
-24
-16
-8 = /3 or /2
5 = 10/2 or 20/4
»» Write 5/2 = 10/4 on the
board.
»» Do students
remember
what a prime
number is?
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Teaching & Learning Plan: Inequalities
Input
Student Activities: Possible and Teacher’s Supports and
Checking
Expected Responses
Actions
Understanding
»» What are the combined Rational • Real Numbers.
Explain clearly that Z is a subset »» Are students aware
and Irrational Numbers called?
of and do they
of R, while the set of Rational
R
»» What letter represents set of
Real Numbers?
•
»» Are Rational Numbers also Real
Numbers?
• Yes
and Irrationals are disjoint.
• Students give examples of Real
Numbers, Rational Numbers
and Irrational Numbers.
»» Can you give an example of
a Real Number that is not a
Rational Number?
• For example √5,√7 or π.
»» Is it possible to represent the
Irrational Numbers accurately
on the number line? Why?
• No, not accurately. We can
make estimates, but as they
are non-repeating decimals
it depends on the degree of
accuracy required.
»» Answer the questions in Section
B: Student Activity 1.
• Students work on Section B:
Student Activity 1.
»» Do students
understand the
diagram that is on the
board to represent
the number systems?
Note: It is possible to use
constructions to locate
irrationals along the number
line. The inherent inaccuracy
then arising from the width of
the pencil and other limitations
associated with constructions
can be explored.
»» Distribute Section B: Student
Activity 1.
»» Circulate the room and offer
assistance where required.
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understand the
different types of
numbers?
»» Do students see that
of a Real Number
that is not a Rational
Number is another
irrational number?
»» Are Irrational Numbers also Real • Yes
Numbers?
»» Give examples of Real Numbers,
Rational Numbers and Irrational
Numbers.
Teacher Reflections
KEY: » next step
»» Can students
distinguish between
the different number
systems?
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Teaching & Learning Plan: Inequalities
Student Activities: Possible Teacher’s Supports and Actions
and Expected Responses
Checking
Understanding
Section C: Number Systems and the Number Line.
»» Where along the number line on the
board does the set of Natural Number
begin?
Does it include 1?
Which direction is the arrow pointing?
As the arrow is on a continuous line,
what set of numbers do you think is
being represented on the following
number line.
»» Draw the number line on the
board.
• 1
Yes.
To the right.
x ≥ 1, x ∈ R
an open circle, how do you think the
inequality would differ?
• It would be greater than
rather than greater than
or equal to.
Note: When solving inequalities it is very
important that the number system to
which the number belongs is stressed.
•
x ≥ 4, x ∈ N
•
x > 1, x ∈ R
•
x ≤ 1, x ∈ R
•
x < 1, x ∈ R
»» What does each of the diagrams on the
board represent?
»» These packages contain a number of
statements, mathematical inequalities
and number lines. For every statement
I want you to match the statement,
with the appropriate mathematical
inequality and number line.
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Teacher Reflections
• Students work in pairs
discussing and comparing
their work.
»» Draw the following diagrams
on the board and discuss their
differences.
»» Distribute packages containing
the activity from Section C:
Student Activity 1.
»» Circulate the room and offer
assistance when required.
KEY: » next step
»» Can students
distinguish
between the
different
inequalities
represented
on these
number
lines?
»» Can students
represent
the
inequalities
correctly on
the number
line?
12
Teaching & Learning Plan: Inequalities
Student Activities: Possible and
Expected Responses
Teacher’s Supports and
Actions
Section D: Solving inequalities of the form ax + b <k.
»» See question 1 Section D: Student Activity 1.
»» Annika has seen three pairs of trainers she
likes. They cost €50, €55 and €68. She already
has saved €20 and gets €4 pocket money
per week at the end of each week. Annika is
wondering how soon she can buy one of these
pairs of trainers. What different methods can
you use to help her solve this problem?
»» Which trainers will she be able to buy first?
»» Now I want you to arrive at a strategy to
solve this problem. I will be asking some of
you to present your strategy to the class and
I will want you to discover the earliest, when
Annika can afford to buy one of these pairs of
trainers.
Checking
Understanding
Teacher Reflections
»» Distribute Section D:
Student Activity 1.
»» Give students time
to come up with
their strategies and
then allow as many
students as possible to
present their strategy,
• €50
writing their tables,
Trial and error: She has €20 and will
graphs, calculation
need another €30 so 30 divided by
etc. on the board.
4 equals 7.5. So after 8 weeks she
Allow all strategies to
will have more than €50 and she
remain on the board if
will be able to buy the trainers.
possible until a solution
• A table
involving an inequality
is produced.
Week
Amount saved
0
€20
1
€24
2
€28
3
€32
4
€36
5
€40
6
€44
7
€48
8
€52
»» Can students
solve the
problem by
trial and
error?
»» Can students
solve the
problem by
the table
method?
»» Students may begin
with a table and then
move to algebra. This is
perfectly acceptable.
»» Students may produce
other strategies, for
example diagrams,
and if so, acknowledge
these strategies.
She can afford the €50 trainers at
the end of week 8.
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Teaching & Learning Plan: Inequalities
Lesson Interaction
Teacher Input
Student Activities: Possible and
Expected Responses
• 20 + 4x, where x is the number
»» Do you see any pattern
forming?
Teacher’s Supports and
Actions
Teacher Reflections
of weeks.
• A graph
Note: It is also worth
discussing that she will
be able to afford the
cheapest pair at any
time after the earliest
date and this underlines
the difference between
an equation and an
inequality.
»» What formula is
• f(x) = 20 + 4x
»» Why did you use a dashed
line?
• Because the data is discrete
»» Now we will try and
put this problem in
mathematical language.
»» In x weeks how much has
she saved in total?
• 4x
»» How much will she have
saved in total after x
weeks?
• 20 + 4x
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Checking Understanding
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»» Can students solve the
problem by means of a
graph?
»» Do students understand
that as the pocket money
is paid at the end of the
week it will be the end of
week 8 before Annika can
afford the trainers and not
half way through week 7
as the graph appears to
indicate?
»» A student, who solved
the problem using
an equation, may be
solution at this stage.
»» Write the inequality on
the board as it evolves
and find its solution.
KEY: » next step
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Teaching & Learning Plan: Inequalities
Input
Student Activities: Possible
and Expected Responses
Teacher’s Supports
and Actions
»» In order to be able to buy the
trainers, what is the minimum
amount she will have to save?
• 50
»» A student, who
solved the problem
using algebra solely,
present their solution
at this stage.
»» Write this using mathematical
symbols.
»» Now remembering how we
solved an equation, can you solve
this inequality?
• 20 + 4x ≥ 50
• Stabilisers and as follows:
-20
÷4
20 + 4x ≥ 50
4x ≥ 30
x ≥ 7.5
-20
÷4
»» When does she get her pocket
money?
• At the end of each week.
»» So when will she be able to
afford the €50 trainers?
• It will be at least 8 weeks
before she can afford these
trainers.
»» Write the inequality
on the board as it
evolves and find its
solution.
Checking Understanding
Teacher Reflections
»» Can students translate
the problem into an
inequality?
»» Can students solve the
inequality?
»» Do students understand
that solving a linear
inequality is similar
to solving a linear
equation?
»» Can students explain
context of the question?
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Teaching & Learning Plan: Inequalities
Teacher Input
Student Activities: Possible and Teacher’s Supports and
Expected Responses
Actions
• Find a value(s) for x that
»» What does it mean to solve
an equation?
Checking Understanding
Teacher Reflections
makes the equation true.
• Find the set of values for x
that make the inequality true.
»» What does it mean to solve
an inequality?
»» How does an equation and
an inequality differ?
• A
n equation uses the = symbol »» Write the differences on the
board.
and an inequality uses one of
the <, >, ≤ or ≥ symbols.
»» Tell the students how it is
An inequality can have more
possible for equations to
than one solution.
have more than one solution
and distinguish this from
»» Solve the following inequality
+3
+3
the solution to an inequality
2x - 3 <7
2x - 3 < 7, x ∈ N. Show your
÷2
÷2
whose solution set contains
2x <10
calculations.
a range of values of x.
x <5
»» List the possible outcomes of
the inequality and show on
the number line.
• {1, 2, 3, 4}
»» Write the possible outcomes
and their representation
on the number line on the
board.
»» If x had been an element
of R (x ∈ R) how would the
solution appear on number
line?
»» Complete questions 2-4 in
Section D: Student Activity 1.
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»» Write the requisite
inequality and calculations
on the board.
• Students work on questions
2-4 in Section D: Student
Activity 1.
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»» Can students see
the significance
of the different
domains (x ∈ N and
x ∈ R) for example,
when representing
the solutions to
inequalities?
»» Circulate the room and offer »» Do students distinguish
assistance when required.
for a possible solution
to an inequality and
the solution?
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Teaching & Learning Plan: Inequalities
Student Learning Tasks: Teacher Student Activities: Possible Teacher’s Supports and
Input
and Expected Responses
Actions
Checking Understanding
Solving inequalities of the form ax + b <cx + d.
»» Read question 1 Section D:
Student Activity 2 in your
worksheet.
John has 18 ten-cent coins in
his wallet and Owen has 22
five-cent coins in his wallet.
Each day, they decide to take
one coin from their wallets
and put it into a money box,
until one of them has no more
coins left in their wallet.
When did Owen have more
money than John in his wallet?
»» Now I want you to arrive at a
strategy to solve this problem.
I will be asking some of you
the class and explain how you
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Teacher Reflections
»» Distribute Section D:
Student Activity 2.
»» In this case get students to
work individually.
»» Give the students time
to individually develop a
strategy.
»» Observe the students’
reactions and strategies.
»» Encourage students to
develop more than one
strategy.
»» Circulate the room and
pick strategies used by
students that you would
like presented on the board.
Note students may present
different strategies or
combinations of strategies
to those listed in this T&L
and it is fine to use those.
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Teaching & Learning Plan: Inequalities
Student Learning
Student Activities: Possible and
• Student 1: I designed a picture
to represent the ten-cent coins
and another set in a different
colour to represent the fivecent coins and acted out the
problem and got day 15 as my
solution.
• Student 2: I made out the
following tables and noticed
after day 14 Owen had more
money than John.
John
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
Owen
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
• At day 14 they had equal
amounts and after that Owen
had more as seen from my
table.
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Teacher’s Supports and Actions
Checking Understanding
»» Now get some of the students
»» Can students explain their
to present to the class and
strategy and justify it?
request the remainder of the
class to record the strategies and »» Did students read the
solutions they observe.
problem correctly?
»» There are probably many
is only on the 15th day. At this
stage do not go into this too
deeply, but have them think
inequalities.
Teacher Reflections
»» Do students understand
the various solutions?
»» Get the students who solved the
problem using Algebra to use
the data in the problem to set
up an inequality.
»» Ask each student who presents
their solution to explain their
solution.
»» Keep each strategy on the board
until you are finished discussing
the problem.
»» Allow other students to pose
questions of the student who is
presenting.
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Teaching & Learning Plan: Inequalities
Student Learning
Student Activities: Possible and
Teacher’s Supports and
Actions
Checking Understanding
• Student 3: I made out the following
table
Day
John
Owen
0
180
110
1
170
105
2
160
100
3
150
95
4
140
90
5
130
85
6
120
80
7
110
75
8
100
70
9
90
65
10
80
60
11
70
55
12
60
50
13
50
45
14
40
40
15
30
35
16
20
30
17
10
25
18
0
20
• From the table I saw that after day
14 Owen had more money than John
and I stopped my table at day 18 as
John’s money ran out at that stage.
Hence Owen has more money than
John on days 15, 16, 17 and 18.
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Teacher Reflections
»» Encourage this student
days 15, 16 17 and 18 and
explain why this is the
solution to our problem.
»» Do students understand
this solution?
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19
Teaching & Learning Plan: Inequalities
Student Activities: Possible and Expected Responses
Student
Teacher Input
• Student 4: I decided that the amount of money in John’s wallet can
be represented by 180 – 10d and the amount in Owen’s wallet can
be represented by 110-5d, where d is the number of days that have
elapsed. I then drew graphs to represent both patterns.
220
Money in wallet
200
180
160
f(x) = 180 - 10d
140
120
100
80
60
40
g(x) = 110 - 5d
A =(14, 40)
20
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Days
Teacher’s Supports
and Actions
Checking
Understanding
Teacher Reflections
»» Discuss why dashed
lines were used in
drawing the graphs
and what the point
of intersection
means. Also
discuss what is the
significance of the x
and y axis intercepts
within the context
of the problem.
The lines representing the patterns met at (14, 40). This was the day
they were equal but the question asked when they were greater.
»» In the case of the
Hence on days 15, 16, 17 and 18 Owen had more money than John.
student who used
an equation to solve
• Student 5: I noticed that the amount of money in John’s box could
the problem, get
be represented by 180-10d and the amount in Owen’s by 110-5d
him/her to present
where d is the number of days elapsed. Then I solved my equation.
this to the class with
an emphasis on
+ 5d
180 - 10d = 110 - 5d + 5d
setting it up rather
- 180
180 - 5d = 110
- 180
than on solving it.
÷ (-5)
-5d = 70
÷ (-5)
»» Do students
understand the
significance of
the point of
intersection of
the graphs?
»» Do they
appreciate that
it is not only day
14 that Owen’s
wallet contains
more money
than John’s?
»» Do students
remember their
methodologies
for solving
equations.
d = 14
This means on day 14 they have equal amounts of money, but
the question asked when has Owen more money than John in his
wallet. Hence on days 15, 16, 17 and 18 Owen has more money
than John in his wallet. I stopped at day 18 as John’s money ran
out on that day and the question stated they stopped when one of
them had no more money in their wallet.
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20
Teaching & Learning Plan: Inequalities
Student Activities: Possible
and Expected Responses
»» Let us examine this table a little further.
»» Insert <, > or = between John and
Owen as appropriate.
»» So, on what day were the amounts in
Owen and John’s wallets equal?
• Day 14.
»» On what days was the amount in
Owen’s wallet greater than that in
John’s?
• Days 15, 16, 17 and 18.
»» How could we represent the amount
of money John has in his wallet on any
particular day algebraically?
• 180 - 10d
»» How could we represent the amount of
money Owen has in his wallet on any
particular day by algebra?
• 110 - 5d
»» Hence how could we represent our
problem algebraically?
• 110 - 5d> 180 - 10d
»» Can we solve this inequality using a
strategy similar to that used for solving
simple equations?
• Stabilisers and as follows:
+5
»» What difficulty occurred if we just used
an equation to solve this problem?
© Project Maths Development Team 2013
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Checking
Understanding
»» Write this on the board
if a suitable table has not
»» Can the students
solve the problem
using this
method?
Day
John
13
50
>
Owen
45
14
40
=
40
15
30
<
35
16
20
<
30
17
10
<
25
18
0
<
20
»» Write the student
and discuss each step as it
appears in the inequality.
+ 10d 110 - 5d> 180 - 10d + 10d
- 110
Teacher’s Supports and
Actions
180 - 5d> 180
5d> 70
d> 14
- 110
+5
Teacher Reflections
»» Can students
represent the
situation using
algebra?
»» Notice that the solution
»» Do students
set of the inequality
understand that
contains a range of values
it is insufficient to
of x.
simply solve the
equation and go
Note: Be careful to ensure
no further?
that the inequality does not
involve -d as students will
not have the skills to deal
with this as yet.
• It only gave us an answer of
the question and see that it
was days 15, 16, 17 and 18.
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21
Teaching & Learning Plan: Inequalities
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Supports and
Actions
Checking Understanding
»» We have just solved an
inequality rather than an
equation.
»» Do students recognise
the differences between
an equation and an
inequality?
»» How do the approaches to
solving linear inequalities
and equations differ? How
are they the same?
• We used the same
method, but with an
inequality we can have a
range of solutions.
»» Complete the remaining
questions in Section D:
Student Activity 2.
• Students work on Section
D: Student Activity 2
comparing and discussing
»» Circulate the room and offer
assistance as required.
»» Working in pairs, complete
the Tarsia that has been
distributed. It is a domino
jigsaw. You will need to
find the piece with Start
written on it. This piece
also contains an algebraic
expression. You must
match each piece with
another piece containing a
corresponding expression.
Continue matching until
you arrive at Finish.
• Working in pairs, students
complete the Tarsia.
»» Distribute Section D:
Student Activity 3.
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Teacher Reflections
»» Can students complete this
Tarsia?
KEY: » next step
22
Teaching & Learning Plan: Inequalities
Student Activities: Possible
and Expected Responses
Teacher’s Supports
and Actions
Checking
Understanding
Section E: Multiplying and Dividing by a Negative Number.
»» Complete the exercise contained in
Section E: Student Activity 1 Question 1.
Teacher Reflections
»» Distribute Section E:
Student Activity 1.
»» Is -3 < 5 true?
• Yes
»» When you multiply -3 “by”-1,”what do
you get?”
• 3
»» When you multiply 5 “by”-1,”what do
you get?”
• -5
»» Is 3 greater or less than -5?
• 3 > -5
»» So what happened when you multiplied
the inequality by -1?
• The direction in the
inequality reversed.
»» Write the various
steps on the board.
»» Do students
recognise that
multiplying (or
dividing) an
inequality by -1
causes the direction
of the inequality to
reverse?
»» In groups of two write down four
inequalities of your choice and multiply
each by -1 and observe the effect in each
case.
»» Repeat the exercise, but multiplying
or dividing through by a few different
negative numbers.
»» What happens in each case?
• < becomes > and
> becomes <.
»» Complete Section E: Student Activity 1.
• Students complete Section
E: Student Activity 1.
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»» Discuss with the class
how ≤ becomes ≥
and ≥ becomes ≤.
KEY: » next step
23
Teaching & Learning Plan: Inequalities
Teacher Input
Student Activities: Possible and
Expected Responses
»» When you multiply both
sides of an inequality by
the same negative number,
what happens to the
inequality?
• The direction of the inequality
reverses.
»» Complete the questions
in Section E: Student
Activities 2 and 3.
• Students work on Section E:
Student Activities 2 and 3 to
consolidate their learning.
»» Now we are going to
complete the following
table as a conclusion to the
previous exercises.
Action (to both sides of an
Does the
inequality)
direction
of the
inequality
change?
Yes / No
No
Subtract a positive number
No
No
Subtract a negative number
No
Teacher’s Supports
and Actions
Checking Understanding
»» Do students understand what
happens to the inequality
when one multiplies both
sides of the inequality by the
same negative number?
Teacher Reflections
»» Distribute Section E:
Student Activities 2
and 3.
»» Complete the titles
and first column of
the table, opposite.
»» Through discussion
with the students
complete the table.
»» Do students understand that
when one multiplies or divides
both sides of the inequality
by a negative number, the
direction of the inequality is
reversed?
Multiply by a positive number No
Multiply by a negative
Yes
number
»» What actions applied to
both sides of an inequality
causes the direction of the
inequality to reverse?
© Project Maths Development Team 2013
Divide by a positive number
No
Divide by a negative number
Yes
• Multiply and divide both sides
of the inequality by a negative
number.
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KEY: » next step
24
Teaching & Learning Plan: Inequalities
Student Activities: Possible
and Expected Responses
Teacher’s Supports
and Actions
Checking
Understanding
Section F: Solving inequalities of the form –ax+b<c.
»» What is the first action you would
perform to solve the inequality
-x + 2 > 5, x ∈ R?
• Subtract 2 from each side.
»» What will the inequality now look like?
• -x > 3
»» What will the next step be?
• Divide both sides by -1.
»» What happens when you divide both
sides of an inequality by any negative
number?
• > becomes < and
< becomes >.
»» So what is the solution of the
inequality?
•
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Teacher Reflections
»» Write each step
»» Do students
involved in solving
recognise that
the inequality on the
this inequality is
board.
different from what
they have met to
date?
»» Do students
recognise that the
direction of the
reverse because the
solution required
division of both
sides by -1?
x < -3, x ∈ R
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25
Teaching & Learning Plan: Inequalities
Student Activities: Possible and
Expected Responses
Teacher’s Supports
and Actions
Checking
Understanding
Section G: Solving inequalities graphically.
»» Complete question 1 in Section G:
Student Activity 1.
x
f(x) = 3x + 6
-3
-3
-2
0
-1
3
0
6
1
9
»» Distribute Section G:
Student Activity 1.
»» Allow students time
to engage with this
question.
8
6
f(x) = 3x + 6
4
2
-3
-2
-1
0
1
x = -2
»» Where does the line cut the x axis?
•
»» When is f (x) > 0?
• It is greater than zero for x
values greater than minus two.
»» When is f (x) < 0?
• It is less than zero for x values
less than minus two.
»» In solving an equation we find the x
values that make it true and in the same
way when we solve an inequality we
find the values of x that make it true.
»» So what x values make the inequality
3x + 6 ≥ 0 true?
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•
x ≥ -2
Teacher Reflections
»» Do students
understand that
when the line is
above the x axis
»» Encourage the more
the values of f(x)
able students to
are positive and
draw the function
when it is below
by locating the x
the x axis they
and y intercepts only
are negative?
and then checking
that other points in
»» Can students
the table satisfy the
solve linear
equation of the line.
inequalities
graphically?
»» After the students
»» Do students
opportunity to
know that the
produce a table and
solutions are the
graph themselves
values on the x
through discussion
axis that make
draw the table and
the inequality
graph on the board.
true?
»» Encourage the
students to check
if this is true using
other values of x.
KEY: » next step
26
Teaching & Learning Plan: Inequalities
Lesson Interaction
Student Learning Tasks: Teacher Student Activities: Possible
Input
and Expected Responses
»» What x values make the
• x ≤ -2
inequality 3x + 6 ≤ 0 true?
»» What x values make the
inequality 3x + 6 > 0 true?
•
x > -2
»» What x values make the
inequality 3x + 6 < 0 true?
•
x<2
»» Complete the exercises in
Section G: Student Activity 1.
• Students complete the
exercises in Section G:
Student Activity 1.
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Teacher’s Supports and
Actions
Checking Understanding
Teacher Reflections
»» Circulate the room and
offer assistance when
required.
KEY: » next step
27
Teaching & Learning Plan: Inequalities
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Supports and
Actions
Checking Understanding
Section H: To investigate the rules of inequalities.
Note: in this exercise a and
b are Real Numbers unless it
states otherwise.
»» Distribute Section H:
Student Activity 1.
»» Working in groups, use
suitable values for a, b, c
to investigate the activities
contained in Section H:
Student Activity 1.
• Working in pairs, students
discuss these inequalities.
Note: Solutions are contained
in this Teaching and Learning
plan on page 52.
»» Encourage students
to replace a, b and
c with positive and
negative whole numbers
and fractions to help
them investigate the
relationships.
»» Circulate the room and
offer assistance when
required.
»» Now I want the selected
students to present their
solutions on the board.
• Selected students present
their solutions on the
board.
»» Can students replace a,
b and c with numerical
values to investigate the
relationships?
»» Do students understand
that these inequalities
have to be true for all
values?
»» Do students understand
+
the impact of c ∈ R and
c ∈ R-?
»» Can students justify their
»» Select students to present
their findings to the class.
»» Encourage the students
who are presenting to
reinforce the validity
substituting with positive
and negative whole
numbers and fractions.
Note: I am particularly
interested in the reasons
behind the solutions.
© Project Maths Development Team 2013
Teacher Reflections
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KEY: » next step
28
Teaching & Learning Plan: Inequalities
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Supports and
Actions
Checking Understanding
Section I: Double inequalities.
Teacher Reflections
»» Angus wants to buy a
present for a friend and he
wants to spend at least €24
and no more than €30.
»» Does this include €24 and
€30?
• Yes
»» What is the least he can
spend?
• €24
»» What is the most he can
spend?
• €30
»» Write this problem as a set
of inequalities?
•
»» What is another way of
writing the inequality
x ≥ 24?
»» Now we can put the two
inequalities together as
24 ≤ x ≤ 30.
x ≥ 24 and x ≤ 30?
• 24 ≤ x
•
x is any Real Number
between 2 and 5, but
cannot be equal to 2 or 5.
»» Write the individual and
combined inequalities on
the board.
»» Do students understand
that x ≥ 24 is equivalent to
24 ≤ x?
»» What does the following
inequality mean
2 < x < 5, x ∈ R?
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KEY: » next step
29
Teaching & Learning Plan: Inequalities
Teacher Input
Student Activities: Possible and
Expected Responses
»» How could we represent
-2 < x < 5, x ∈ R on the
number line?
»» Why did you use open
circles at -2 and at 5?
• Because those points are not
included, it is between -2 and 5.
been ≤ instead of <, how
would this affect the
representation of the
inequality on the number
line?
• It would have been closed circles
as -2 and 5 would have been
included.
Teacher’s Supports and
Actions
Checking Understanding
»» Draw the number line on
the board and write the
inequality it represents
beside it.
»» Do students understand
how to represent
inequalities of the form
a < x < b, for different
number systems on the
number line?
»» Draw a diagram on the
board to illustrate this
inequality.
Teacher Reflections
»» Distribute Section I:
Student Activity 1.
»» If the inequality had been
• You would just have put closed
-2 < x < 5, x ∈ Z, what
circles at the relevant points -1,
0, 1, 2, 3 and 4
»» Do the exercises 1-5
contained in Section I:
Student Activity 1.
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KEY: » next step
30
Teaching & Learning Plan: Inequalities
»» How would one solve the inequality
3 ≤ - 2x + 1 ≤ 7, x ∈ R?
Student Activities: Possible
and Expected Responses
-1
÷2
3 ≤ -2x + 1 ≤ 7
2 ≤ -2x ≤ 6
1≤-x≤3
Teacher’s Supports and
Actions
-1
÷2
Divide by -1 gives -1 ≥ x ≥ -3.
»» Now write this inequality as two
inequalities.
• -1 ≥ x and x ≥ -3
»» Can any of these inequalities be
simplified?
• Yes. The first can be written
as x ≤ -1.
»» On the board, draw a
diagram to illustrate
that the solution is more
correctly represented as
-3 < -1.
Checking
Understanding
»» Can students
solve and
represent the
solutions of
inequalities
of the form
a < x < b on the
number line?
Teacher Reflections
»» Now draw this on the number line.
»» So what is another way of writing the
solution?
• Another way of writing this
inequality is -3 ≤ x ≤ -1.
»» Complete the remaining exercises in
Section I: Student Activity 1.
»» Working in pairs complete the Tarsia
• In pairs students complete
that has been distributed. It is a
the Tarsia.
domino jigsaw. You will need to find
the piece containing the word Start. It
also contains an algebraic expression;
you must find the equivalent
expression in another piece and you
continue matching until you arrive at
Finish.
»» Distribute Section I:
Student Activity 2.
»» Match the appropriate words, sample
sentences, equivalent forms and
translations.
»» Distribute Section I:
Student Activity 3.
© Project Maths Development Team 2013
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• In pairs students complete
the graph matching exercise.
KEY: » next step
»» Can students
complete the
Tarsia?
31
Teaching & Learning Plan: Inequalities
Section A: Student Activity 1
Revision of < and > symbols.
1. The table below contains a number of inequalities. In the space provided,
indicate which are true and which are false
2<3
-1 > 4
-2 > -1
4≤5
1 > -4
3>4
-1 ≤ 4
1.2 < 4
-1 ≥ 4
-1.8 > 4
1
/2 < 3/4
1
-1/2 < 3/4
1
/2 > 1/4
/2 > -1/4
2. Insert the appropriate symbol between these numbers.
Insert < or > between these numbers
6
-6
5
1.5
-6
10
-10
-4
3.5
-4
1
1
/2
/4
-1/2
20%
-1/4
0.02
3. In each case, below, circle the algebraic expression which represents the
statement given.
a
is less than 5
x > 5
x < 5
x ≤ 5
x≥5
b
is more than 8
x > 8
x < 8
x ≤ 8
x≥8
c
is less than or equal to 4
x > 4
x < 4
x ≤ 4
x≥4
d
is greater than or equal to 10
x > 10
x < 10
x ≤ 10
x ≥ 10
e
is at least 10
x < 10
x > 10
x ≥ 10
x ≤ 10
f
is at most 10
x < 10
x > 10
x ≤ 10
x ≥ 10
g
Let r be the amount of rain (in mm) which falls each day. More than 23mm of rain fell yesterday.
r < 23
r > 23
r ≤ 23
r ≥ 23
h
p is no more than 9
© Project Maths Development Team 2013
p < 9
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p > 9
p ≥ 9
p≤9
32
Teaching & Learning Plan: Inequalities
Section A: Student Activity 1
4. For each of the statements below, circle the inequality which represents the given
statement.
a
The speed limit on a certain road is 60 km. Does this mean each driver has to drive at this speed or
less? s= speed in km/h. Represent the speed limit using the variable s.
s < 60 s ≤ 60 s > 60 s ≥ 60
b
In order to be able to go on a school trip Kelly needs to have saved €40 or more. Kelly has saved €d
and is not yet able to go on the trip. Which of the following is true?
d ≤ 40
d ≥ 40
d > 40
d < 40
c
To enter a particular art competition you must be at least 12 years old. Tom is n years old and he can
enter the competition. Which of the following is true?
n < 12
n ≤ 12
n ≥ 12
n > 12
d
To enter an art competition you must be over 12 years old. Tom is n years old and he can enter the
competition. Which of the following is true?
n < 12
n ≤ 12
n ≥ 12
n > 12
e
The maximum number of people allowed in a cinema is 130. If there are b people in the cinema.
Which of the following is true?
b > 130
b ≥ 130
b ≤ 130
b < 130
f
The best paid workers in a business earn €40 per hour. Mark earns €m per hour and he is not one of
the best paid employees in the business. Which of the following is true?
m ≥ 40
m < 40
m > 40
m ≤ 40
g
Emma’s mother says that when she reaches the age of 17 she will get an increase in her pocket
money. Emma is r years old and has not yet received that increase. Which of the following is true?
r < 17
r > 17
r ≤ 17
r ≥ 17
h
There are at least 200 animals in a zoo. If there are h animals in the zoo, which of the following is
true?
h > 200
h < 200
h ≥ 200
h ≤2 00
i
At most 10 people can fit in a bus and there are k people in the bus. Which of the following is true?
k < 10
k > 10 k ≤ 10
k ≥ 10
j
A film is given an age 18 certificate. Let a = age of a student. To watch the film which statement is
true?
a < 18
a > 18
a ≤ 18
a ≥ 18
k
In Ireland you have to be at least 18 years old in order to be able to vote. Conor is w years old and
he can vote, Which of the following is true?
w ≤ 18
w < 18
w > 18
w ≥ 18
l
In order to get to the next stage of the competition a team must have at least 20 points. Given x =
the number of points scored by a team and they qualify for the next stage of the competition, which
of the following is true?
x < 20 x ≤ 20 x > 20 x ≥ 20
m
The temperature in Dublin on a particular day is 20°C and it is warmer in Cork on the same day.
Given x°C is the temperature in Cork, which of the following is true?
x ≤ 20
x < 20
x > 20
x ≥ 20
© Project Maths Development Team 2013
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33
Teaching & Learning Plan: Inequalities
Section B: Student Activity 1
Revision of Number Systems.
1. Write the relevant numbers from below into each of the boxes A, B, C, E, F, G
and H. Note: Numbers may be used more than once.
-3, 0∙5, 6, 8, 10, -4, 20, -5, 3½, -6∙2, 1, 11, 2, 3, 9, 5, 5∙2, -2, 7∙9, 12, 11∙3.
A: A Natural Number greater than 2
B: A Real Number greater than 2
C: A Natural Number less than 9
D: An Integer less than 9
E: A Real Number greater than 7
F: A Real Number greater than or
equal to 9
G: A Real Number bigger than -4 and
less than or equal to 5
H: An Integer greater than -2
2. Write down one number which you included in box B, but did not include in
box A. Give a reason for your choice._____________________________________
3. x > 5, x ∈ N means: (Note: There may be more than one correct answer.)
(i) a number less than 5 is a possible solution
(ii) 5 is a possible solution
(iii) 6 is a possible solution
(iv) 5 is the only solution
(v) natural numbers should be Natural Numbers
(vi)Every natural number greater than 5 represents a general solution
4. x > 5, x ∈ N means: (Note: There may be more than one correct answer.)
(i) a number less than 4 is possible solution
(ii) 8 is a possible solution
(iii) 6 is a possible solution
(iv) 5 is the only solution
© Project Maths Development Team 2013
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34
Teaching & Learning Plan: Inequalities
Section B: Student Activity 1
5. What is the general solution to the inequality in Q4?
6. x ≤ 6, x ∈ N State giving a reason which of the following are true or false.
i. 6 is a possible solution_________________________________________________
ii. 7∙8 is a possible solution________________________________________________
iii. -4 is a possible solution_________________________________________________
iv. 9 is a possible solution_________________________________________________
v. The solution set contains 6 elements
7.Given x ≤ 9, x ∈ Z, write “True” or “False” beside each of the following.
i. 6 is a possible solution_________________________________________________
ii. 10∙8 is a possible solution______________________________________________
iii. -4 is a possible solution_________________________________________________
iv. 9 is a possible solution_________________________________________________
8.Given x < 8, x ∈ Z, write “True” or “False” beside each of the following.
i. x is less than or equal to 8______________________________________________
ii. x is greater than 8_____________________________________________________
iii. x is less than 8_________________________________________________________
iv. x is 8_________________________________________________________________
v. x can be a fraction_____________________________________________________
9. Write the sentence: “x is a Natural Number less than 4.”, in algebraic form
(using symbols).
10.Write the sentence: “x is a Natural Number greater than 3.”, in algebraic
form.
11.Write the sentence: “y is a Rational Number greater than or equal to 12.”, in
algebraic form.
12.Write the sentence: “p is an Integer less than 7.”, in algebraic form.
13.Which of the statements below represents the pattern 14, 15, 16, 17, 18, 19, ...
(a) x < 14, x ∈ Z
(d) x ≥ 14, x ∈ N
(b) x > 14, x ∈ R
(e) x > 14, x ∈ N
(c) x ≤ 14, x ∈ R
14.Which of the statements below represents the set A? A={-10,-9,-8,-7,-6,-5,…}
(a) x < -10, x ∈ Z
(d) x > -10, x ∈ Z
© Project Maths Development Team 2013
(b) x ≥ -10, x ∈ Z
(e) x ≥ -10, x ∈ R.
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(c) x ≤-10, x ∈ R
35
Teaching & Learning Plan: Inequalities
Section C: Student Activity 1
Number Systems and the Number Line.
A class set of the following should be laminated and cut up prior to the
beginning of the lesson.
x <3, x ∈ N
x <3, x ∈ Z
x <3,x ∈ R
x >3, x ∈ N
x >3, x ∈ Z
x >3, x ∈ R
x ≤3, x ∈ N
x ≤3, x ∈ Z
x ≤3, x ∈ R
x is less than
3 and x is an
element of N
x is less than
3 and x is an
element of Z
x is less than
3 and x is an
element of R
x is greater
than 3 and x
is an element
of N
-1
0
1
2
3
4
6
5
7
x is greater
than 3 and x
is an element
of Z
x is greater
than 3 and x
is an element
of R
x is less than
or equal to
3 and x is an
element of N
x is less than
or equal to
3 and x is an
element of Z
x is less than
or equal to
3 and x is an
element of R
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-1
0
1
2
3
4
5
6
7
-3
-2
-1
0
1
2
3
4
5
-2
-3
-1
0
1
2
3
4
5
-2
-1
0
1
2
3
4
5
-2
-1
0
1
2
3
4
5
6
36
Teaching & Learning Plan: Inequalities
Section C: Student Activity 1
x ≥3, x ∈ N
x ≥3, x ∈ Z
x ≥3, x ∈ R
x is greater
than or equal
to 3 and x is
an element of
-2
-1
0
1
2
3
4
5
6
7
-2
-1
0
1
2
3
4
5
6
7
N
x is greater
than or equal
to 3 and x is
an element
of Z
x is greater
than or equal
to 3 and x is
an element of
-4
R
x <-3, x ∈ Z x is less than
-3 and x is an
element of Z
x <-3, x ∈ R x is less than
-3 and x is an
element of R
x ≤-3, x ∈ Z x is less than
or equal to
-3 and x is an
element of Z
x ≤-3, x ∈ R x is less than
or equal to
-3 and x is an
element of R
x ≥-3, x ∈ Z x is greater
than or equal
to -3 and x is
an element
of Z
x ≥-3, x ∈ R x is greater
than or equal
to -3 and x is
an element of
-6
-5
-3
-2
-1
0
1
2
3
4
5
6
-7
-6
-5
-4
-3
-2
-1
0
1
2
-5
-4
-3
-2
-1
0
1
2
3
4
-7
-6
-5
-4
-3
-2
-1
0
1
2
-6
-5
-4
-3
-2
-1
0
1
2
3
-4
-3
-2
-1
0
1
2
-4
-3
-2
-1
0
1
2
3
4
R
© Project Maths Development Team 2013
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37
5
Teaching & Learning Plan: Inequalities
Section D: Student Activity 1
Solving inequalities of the form ax + b <k.
1. Annika has seen three pairs of trainers she likes. They cost €50, €55 and €68.
She already has saved €20 and gets €4 pocket money per week at the end of
each week. Annika is wondering how soon she can buy one of these pairs of
trainers. What different methods can you use to help her solve this problem?
2. Which of the following is an element of the solution set of the inequality
4x + 5 > 29?
i. 4
ii. -5
iii. 6
iv. 7.
3. Solve the following inequalities for x ∈ R and represent their solutions on the
number line:
i. x + 3 <5
ii. x -6 <5
iii. 2x + 3 <5
iv. 2x + 3 <15
v. 12x + 3≥ 111
4. Complete the following two tasks for each of the problems below:
i. Solve the problem using a table, graph or trial and error. Show your
calculations and explain your reasoning in all cases.
ii. Form an inequality to represent the problem and solve it algebraically.
a. Darren worked a six hour shift in his local restaurant and got €5 in tips. His
total take home pay that evening was at least €69, find the minimum amount
he was paid per hour.
b. A farmer wants to buy some cows and a tractor. The tractor costs €20,000 and
the maximum he can spend is €60,000. Given that the average price of a cow is
€900, find the maximum number of cows he can buy.
c. Ronan buys a tomato plant which is 5 centimetres in height. The tomato plant
grows 3 centimetres every day. After how many days will the tomato plant
reach the top of the glass house in which it is growing, given the glass house is
2 metres high.
d. Declan is saving for a birthday present for his girlfriend and he already has €6.
Given that he plans to spend at least €40 on the present and the birthday is
five weeks hence, what is the least amount he should save per week?
e A bridge across the river Geo can support a maximum weight of 20 tonnes. The
company’s lorry weighs 8 tonnes and draws a trailer of 3 tonnes in weight. The
company wants to find the maximum cargo they can take across this bridge
using their lorry. Represent the problem, as an inequality and hence find the
maximum weight of the cargo the lorry could carry cross the bridge.
© Project Maths Development Team 2013
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38
Teaching & Learning Plan: Inequalities
Section D: Student Activity 2
Solving inequalities of the form ax + b < cx + d.
1. John has 18 ten-cent coins in his wallet and Owen has 22
five-cent coins in his wallet. Each day, they decide to take
one coin from their wallets and put it into a money box,
until one of them has no more coins left in their wallet.
When does Owen have more money than John in his
wallet?
John
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Owen
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39
Teaching & Learning Plan: Inequalities
Section D: Student Activity 2
2. Declan is having a party and is buying pizzas and chips. He has at most €20 to
spend on food and his budget for chips is exactly €4.60. What is the maximum
number of pizzas he can buy, if each pizza costs €3.50? The shop only sells
whole pizzas.
3. John and Michael go running in order to keep fit. John runs each day from
Monday to Friday and then runs 2 kilometres each Saturday.
Michael goes running on Mondays, Tuesdays and Wednesdays and also runs 9
kilometres each Sunday.
They each run the same fixed distance on the weekdays on which they run.
i. John runs further each week than Michael. Write an inequality to
represent this situation.
ii. Investigate what is the minimum number of kilometres for which this
inequality is true using a table, graph and algebra.
4. When on holiday in France for a week, Emma’s dad hired a car. He paid a fixed
rental of €200 per week and €0.15 per kilometre for each journey undertaken.
He has at most €300 to spend on car hire. What is the maximum number of
kilometres he can drive in the hired car?
5. The length of Lisa's rectangular dining room is 4 metres. If the area of the
room is at least 12 square metres, what is the smallest width the room could
have?
6. Tell the story of possible shopping trips that could be represented by the
following inequalities:
i.3x + 7 < 40
ii.5x + 30 > 50
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40
Teaching & Learning Plan: Inequalities
Section D: Student Activity 3
Tarsia
Cut along the line that separates each row and then cut each section in half
again. This will form a set of dominos. Instruct the students to find the domino
with Start on it and then search for the matching section that accompanies the
Start domino. Continue this until Finish is reached.
2 x > 10
3x + 4 <2x + 3
4x - 6 > 24
x < 12
3(x - 2) > 18
4 < -5
x < -1
x is greater
than or
equal to 5
3x - 4 < 5
x>5
x+7<8
-3 < -2
x-5<7
x is
less than 7
x < -4
x>8
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41
Teaching & Learning Plan: Inequalities
Section D: Student Activity 3 (continued)
x≥5
4x - 10 < -26
True
3x < 15
x>2
x is
at least 7
x<7
x<1
False
x > 4.5
x≥7
Finish
x<5
3x - 8 < -2
Start
x<3
Note: The solution is on the page 43.
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42
Teaching & Learning Plan: Inequalities
Solution to Tarsia
Start
x<3
x is
greater
than or
equal to 5
x < -1
3x + 4
<2x + 3
2 x > 10
x>5
3x - 4
<5
x>8
3(x - 2) >
18
4 < -5
False
x > 4.5
x≥5
4x - 10 <
-26
x < -4
4x - 6 >
24
x < 12
-3 < -2
x+7<8
x<1
3x - 8 <
-2
x>2
x<7
x is
less than
7
x-5<7
x≥7
Finish
True
3x < 15
x<5
© Project Maths Development Team 2013
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x is
at least 7
43
Teaching & Learning Plan: Inequalities
Section E: Student Activity 1
Multiplying and Dividing an inequality by a Negative Number.
1a.Circle the numbers - 3 and 5 on the number line below.
b. Which of these two numbers is smaller? Explain your answer referring to
the number line.
c. Insert the correct symbol (<, >, ≤, ≥) into the box
-3
5.
d. By multiplying the -3 and the 5 by -1 fill in the boxes below and represent
the answer on the number line in part a.
-3 multiplied by -1 =
5 multiplied by -1 =
e Which of the numbers from part d. is the smaller?
2
Original inequality
Multiply by -1
i
4>1
Multiply by -1
ii
3<5
Multiply by -1
iii
6 > -4
Multiply by -1
iv
5≥4
Multiply by -1
v
-5 ≤ 2
Multiply by -2
vi
-1 ≥ -6
Multiply by -3
vii
-8 ≤ -3
Multiply by -1
viii
7≥7
Multiply by -1
ix
5>4
Multiply by -2
x
-6 < -5
Multiply by -5
xi
-4 < 3
Multiply by -1
xii
1
Multiply by -1
⁄4 < 1⁄2
Resulting Inequality
3. From your experience of the exercise above, can you conclude what
happens to an inequality when you multiply both sides by -1?
4. What conclusion do you arrive at when you multiply both sides of an
inequality by the same negative number?
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44
Teaching & Learning Plan: Inequalities
Section E: Student Activity 2
Perform the action contained in the arrow to each side of the given inequality in
the centre.
In which of the new inequalities did the direction of the inequality differ from
the original inequality?
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45
Teaching & Learning Plan: Inequalities
Section E: Student Activity 3
1a
Inequality
2<3
6>4
2=2
a<b
a>b
Action to each side of Result
the inequality
Did you have to reverse the direction of
the inequality?
b If you add the same positive number to each side of the inequality do you
reverse the direction of the inequality? Give examples.______________________
_________________________________________________________________________
_________________________________________________________________________
2a
Inequality
2<5
6>2
7=7
a<b
a>b
Action to each side of
Result
the inequality
Did you have to reverse the direction of
the inequality?
b If you add the same negative number to each side of an inequality do you
reverse the direction of the inequality? Give examples.______________________
_________________________________________________________________________
_________________________________________________________________________
3a
Inequality
3<5
8>2
7 =7
a<b
a>b
Action to each side of the
inequality
Subtract 2
Subtract 4
Subtract 6
Subtract a positive number
Subtract a positive number
Result
Did you have to reverse the
direction of the inequality?
b If you subtract the same positive number from each side of an inequality do
you reverse the direction of the inequality? Give examples.__________________
_________________________________________________________________________
_________________________________________________________________________
© Project Maths Development Team 2013
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46
Teaching & Learning Plan: Inequalities
Section E: Student Activity 3
4a
Inequality
1<6
5>3
4=4
a<b
a>b
Action to each side of the
inequality
Subtract -5
Subtract -3
Subtract -4
Subtract a negative number
Subtract a negative number
Result
Did you have to reverse the
direction of the inequality?
b If you subtract the same negative number from each side of an inequality do
you reverse the direction of the inequality? Give examples.__________________
_________________________________________________________________________
_________________________________________________________________________
5a
Inequality
3<8
9>4
6=6
a<b
a>b
Action to each side of the
Result
inequality
Multiply by 2
Multiply by 3
Multiply by 5
Multiply by a positive number
Multiply by a positive number
Did you have to reverse the
direction of the inequality?
b If you multiply each side of an inequality by the same positive number do you
reverse the direction of the inequality?_____________________________________
_________________________________________________________________________
_________________________________________________________________________
3a
Inequality Action to each side of the
Result
inequality
4<8
Multiply by -2
9>3
Multiply by -4
10 = 10
Multiply by -5
a<b
Multiply by a negative number
a>b
Multiply by a negative number
Did you have to reverse the
direction of the inequality?
b If you multiply each side of an inequality by the same negative number do you
reverse the direction of the inequality?_____________________________________
_________________________________________________________________________
_________________________________________________________________________
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47
Teaching & Learning Plan: Inequalities
Section E: Student Activity 3
7a
Inequality
6 < 18
12 > 6
20 = 20
a<b
a>b
Action to each side of the
Result
inequality
Divide by 6
Divide by 3
Divide by 5
Divide by a positive number
Divide by a positive number
Did you have to reverse the
direction of the inequality?
b If you divide each side of an inequality by the same positive number do you
reverse the direction of the inequality? Give examples.______________________
_________________________________________________________________________
_________________________________________________________________________
8a
Inequality
6 < 10
12 > 4
15 = 15
a<b
a>b
Action to each side of the
Result
inequality
Divide by -2
Divide by -4
Divide by -5
Divide by a negative number
Divide by a negative number
Did you have to reverse the
direction of the inequality?
b If you divide each side of an inequality by the same negative number do you
reverse the direction of the inequality? Give examples.______________________
_________________________________________________________________________
_________________________________________________________________________
© Project Maths Development Team 2013
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48
Teaching & Learning Plan: Inequalities
Section F: Student Activity 1
To solve inequalities of the form -ax + b < c.
1 Solve the following inequalities where x ∈ R:
i. -x + 2 > 5_____________________________________________________________
______________________________________________________________________
______________________________________________________________________
ii. -x + 2 > -5_____________________________________________________________
______________________________________________________________________
______________________________________________________________________
iii. -x -6 < 5______________________________________________________________
______________________________________________________________________
______________________________________________________________________
iv. -2x + 3 < 5____________________________________________________________
______________________________________________________________________
______________________________________________________________________
v. -2x + 3 < 27___________________________________________________________
______________________________________________________________________
______________________________________________________________________
vi. -12x + 3 ≥ 111_________________________________________________________
______________________________________________________________________
______________________________________________________________________
vii.4 - 3x ≥ 46____________________________________________________________
______________________________________________________________________
______________________________________________________________________
viii.-2x + 3 ≥ 10___________________________________________________________
______________________________________________________________________
______________________________________________________________________
ix. -3x + 3 ≥ 17___________________________________________________________
______________________________________________________________________
______________________________________________________________________
2 Fiona has €500 in her bank account. The only payment she makes from this
account is to cover her mobile phone bill of €40 per month. This money is paid
from the bank account at the end of each month. The bank stated they will
warn her when the account goes below €25. Write an inequality to represent
this problem and solve the inequality.______________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
© Project Maths Development Team 2013
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49
Teaching & Learning Plan: Inequalities
Section G: Student Activity 1
Solving inequalities graphically.
1. Complete a table and draw the graph of the function f(x) = 3x + 6, x ∈ R, in
the domain -3 ≤ x ≤ 1 and graphically determine the solution set to each of
the following inequalities:
i. 3x + 6 ≥ 0
ii. 3x + 6 ≤ 0
iii. 3x + 6> 0
iv. 3x + 6 < 0.
2. Complete a table and draw the graph of the function f(x) = x + 4, x ∈ R, in
the domain -3 ≤ x ≤ 3 and graphically determine solution set to each of the
following inequalities:
i. x + 4 ≥ 0
ii. x + 4 ≤ 0
iii. x + 4 > 0
iv. x + 4 < 0.
3. Plot the function f(x) = 2x + 5 and the function g(x) = 5 where x ∈ R on the
same axes and scale. Find the point of intersection of the two graphs and
hence find the solution set of 2x + 3 ≥ 5.
4. Draw the function f(x) = x + 4 and the function g(x) = 2x + 1 where x ∈ R on
the same graph. By examining the graph determine which values of x satisfy
the inequality 2x + 1 < x + 4.
5. Find the equation of the line represented in the diagram below and hence
find an inequality of the form ax + b < k, whose solution is represented by the
arrow on the x axis.
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50
Teaching & Learning Plan: Inequalities
Section H: Student Activity 1
To investigate the rules of inequalities.
Note in this exercise a and b are Real numbers unless it states otherwise.
1 aIf a<b, what do we know about a+c and b+c,if c ∈ R?
bIf a>b, what do we know about a+c and b+c,if c ∈ R?
cIf a<b, what do we know about a-c and b-c,if c ∈ R?
dIf a>b, what do we know about a-c and b-c,if c ∈ R?
2 aIf a<b, what do we know about ac and bc,if c ∈ R+ (Positive Reals)?
bIf a>b, what do we know about ac and bc,if c ∈ R+ (Positive Reals)?
cIf a<b, what do we know about ac and bc,if c ∈ R- (Negative Reals)?
dIf a>b, what do we know about ac and bc,if c ∈ R- (Negative Reals)?
3 aIf a<b, what do we know about a/c and b/c,if c ∈ R+ (Positive Reals)?
bIf a>b, what do we know about a/c and b/c,if c ∈ R+ (Positive Reals)?
cIf a<b, what do we know about a/c and b/c,if c ∈ R- (Negative Reals)?
dIf a>b, what do we know about a/c and b/c,if c ∈ R- (Negative Reals)?
4 a Can one have an expression such as -b> 0? b ∈ R+ Why?
b Can one have an expression such as -b> 0? b ∈ R- Why?
cIf a<b then what is the relationship between 1/a and 1/b?
dIf a>b, how are -a and -b related?
5 aIf a<b and b<c, what do we know about a and c? (Transitive Property)
bIf a>b and b>c, what do we know about a and c? (Transitive Property)
c If Eoin is younger than John and Jean is younger than Eoin, what is the
relationship between John and Jean’s age? (Transitive property)
6 aIf a<0, create an inequality relating a and a2?
bIf a<0, create an inequality relating a and a3?
cIf a<0, create an inequality relating a and a4?
d If 0<a<1, create an inequality relating a and a2?
e If 0<a<1, create an inequality relating a and a3?
f If 0<a<1, create an inequality relating a and a4?
gIf a>1, create an inequality relating a and a2?
hIf a>1, create an inequality relating a and a3?
iIf a>1, create an inequality relating a and a4?
© Project Maths Development Team 2013
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51
Teaching & Learning Plan: Inequalities
Section H: Student Activity 1
Encourage students to support their answers with numerical values in all cases.
1a
1b
1c
1d
2a
2b
2c
2d
3a
3b
3c
3d
4a
If a<b, what do we know about a+c and b+c, if c ∈ R?
If a>b, what do we know about a+c and b+c, if c ∈ R?
If a<b, what do we know about a-c and b-c, if c ∈ R?
If a>b, what do we know about a-c and b-c if c ∈ R?
If a<b, what do we know about ac and bc, if a and b ∈ R and c ∈
R+(Positive Reals)?
If a>b, what do we know about ac and bc, if c ∈ R+ (Positive Reals)?
If a<b, what do we know about ac and bc, if c ∈ R- (Negative Reals)?
If a>b, what do we know about ac and bc, if c ∈ R- (Negative Reals)?
If a<b, what do we know about a/c and b/c, if c ∈ R+ (Positive Reals)?
If a>b, what do we know about a/c and b/c, if c ∈ R+ (Positive Reals)?
If a<b, what do we know about a/c and b/c, if c ∈ R- (Negative Reals)?
If a>b, what do we know about a/c and b/c, if c ∈ R- (Negative Reals)?
Can one have an expression such as -b > 0? b ∈ R+ Why?
4b Can one have an expression such as -b>0? b ∈ R- Why?
4c If a<b then what is the relationship between 1/a and 1/b?
4d If a>b, how are -a and -b related?
5a If <b and b<c, what do we know about a and c?
5b If a>b and b>c, what do we know about a and c?
5c If Eoin is younger than John and Jean is younger than Eoin, what is the
relationship between John and Jean’s age? (Transitive property)
6a
6b
6c
6d
6e
6f
6g
6h
6i
If a<0, create an inequality relating a and a2?
If a<0, create an inequality relating a and a3?
If a<0, create an inequality relating a and a4?
If 0<a<1, create an inequality relating a and a2?
If 0<a<1, create an inequality relating a and a3?
If 0<a<1, create an inequality relating a and a4?
If a>1, create an inequality relating a and a2?
If a>1, create an inequality relating a and a3?
If a>1, create an inequality relating a and a4?
© Project Maths Development Team 2013
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a+c<b+c
a+b>b+c
a-c<b-c
a-c>b-c
ac < bc
ac > bc
ac > bc
ac < bc
a/c < b/c
a/c > b/c
a/c > b/c
a/c < b/c
No. Negative numbers
are always less than
zero.
Yes. Minus a negative
number is always
positive.
1/a>1/b
-a < -b
a < c This is known as
the Transitive Property.
a > c This is also the
Transitive Property.
Jean is younger than
John. John is older than
Jean
a < a2
a > a3
a < a4
a > a2
a > a3
a > a4
a < a2
a < a3
a < a4
52
Teaching & Learning Plan: Inequalities
Section I: Student Activity 1
Double Inequalities
1(Revision)
aRepresent x ≤ 6, x ∈ R on the number line.
bRepresent x ≥ 2, x ∈ R on the number line.
c Represent the intersection of the above two inequalities above on the
number line.
d Rewrite the inequality x≥ 2 in the format a ≤ x.
e Now write the intersection of the two inequalities in the format a ≤ x ≤ b.
2 Maureen takes between two and three hours to do her homework. Write this
as an inequality.
3 A sweet company produces bags of sweets that will contain between 18 and
22 sweets. If there are less than 18 sweets or more than 22 sweets in a bag
then their machine is faulty. Write an inequality to represent the number of
sweets in an acceptable bag of sweets.
4 A farmer says the minimum number of cows he can keep on his farm is 22 and
the maximum is 35, write this as an inequality.
5 Represent the following on the number lines provided:
a 1 ≤ x ≤ 4, x ∈ N
b -3 ≤ x ≤ 4, x ∈ Z
c -3 ≤ x ≤ 4, x ∈ R
d -3 < x ≤ 4, x ∈ Z
e -3 < x < 4, x ∈ Z
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53
Teaching & Learning Plan: Inequalities
Section I: Student Activity 1
f -3 < x < 4, x ∈ R
g -3 ≤ x < 4, x ∈ R
6 Solve the following inequalities and represent the solution set on the number
lines:
a -3 < x + 2 < 5, x ∈ R
b -3 ≤ x + 2 ≤ 5, x ∈ Z
c -2 ≤ x + 3 ≤ 5, x ∈ R
d -2 < x + 3 < 4, x ∈ Z
e -2 ≤ x - 1 ≤ 4, x ∈ R
7 Solve the following inequalities and represent the solution set on the number
line:
a -1 ≤ 2x + 3 ≤ 7, x ∈ R
b -3 ≤2x + 4 ≤ 3, x ∈ R
c 4 < 2x + 1 < 5, x ∈ R
d 4 < 2x + 1 < 5, x ∈ Z
e 4 < 2x + 1 < 5, x ∈ N
f 3 < - 2x + 1 < 5, x ∈ R
© Project Maths Development Team 2013
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54
Teaching & Learning Plan: Inequalities
Section I: Student Activity 1
8 The temperature in a certain town ranged between 18°C and 22°C on a
particular day. Represent this as an inequality in the form a < t < b, where t
represents temperature.
9 Ryan wants to spend between €45 and €67 on a present. If €a is the amount
he wishes to spend, represent his situation as an inequality.
10 The number of matches in a box is between 95 and 103. Represent this
statement as an inequality.
11 Brendan has €x and we know that if he had 3 less it, he would have between
€2 and €10. Represent this as an inequality and solve it.
12 A student wants to keep the cost of his phone calls per week between €5 and
€7 per week. Calls cost €0.20 per call. Write an inequality to represent this and
solve the inequality to determine the maximum number of calls this student
can make while remaining within his budget.
13 A household wants to keep its electricity charges between €30 and €35 per
week. Assuming that there is a standing charge of €5 per week and each unit
of electricity costs €0.10 per unit. Write an inequality to represent the range
of units that this family can use per week.
14 The average number of sweets in a box of a particular brand of sweets is 150,
but the number can vary by ±5. Write an inequality to represent the number
of sweets in the box.
15 A dog food company has a tolerance level of 0.8 Kg on an 8Kg bag of dog
food. Write an inequality that represents an acceptable weight for a bag of
this dog food.
16 Solve 2 (x + 3) > x + 3.
17 Solve -8 < -3x + 1 < 4, x ∈ R and show the solution on the number line.
18 Solve 11 < -4x + 3 < 15, x ∈ R and show the solution on the number line.
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55
Teaching & Learning Plan: Inequalities
Section I: Student Activity 2
Tarsia
Cut along the line that separates each row and then cut each section a half again. This
will form a set of dominos. Instruct the students to find the domino with Start on it and
then search for the matching section that accompanies the Start domino. Continue this
until Finish is reached. Note: The solution is on the page 57.
x ≤ -4
4 < 2x < 6
x>8
x<2
-2 <x <3, x∈Z x is at least 4
x≥4
Finish
-1 < x + 2 < 4
-x < -8
-x > 4
x > -2
2x + 3> 3x +1
x < -4
2<x<3
-3 < x < 2
4 ≤ 2x ≤ 8
{-1, 0, 1, 2}
x+3<6
2≤x≤4
Start
-x + 1 ≥ 5
© Project Maths Development Team 2013
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-3x < 6
x<3
56
Teaching & Learning Plan: Inequalities
Solution to Tarsia
Solution to Tarsia contained in Section I: Student Activity 2 page 56
Start
-x + 1 ≥ 5
-x < -8
-1 < x + 2 -3 < x < 2 2 < x < 3 4 < 2x < 6
<4
x ≤ -4
x>8
x<2
2x + 3>
3x +1
x < -4
-x > 4
x > -2
-3x < 6
x<3
x+3<6
2≤x≤4
Finish
x≥4
© Project Maths Development Team 2013
x is at
least 4
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-2 <x <3, {-1, 0, 1, 2} 4 ≤ 2x ≤ 8
x∈Z
57
Teaching & Learning Plan: Inequalities
Section I: Student Activity 3
Graph Matching Exercise.
These need to be laminated and cut into separate pieces before the class.
Students will then be asked to match them. Note every Important Word does not
have an Equivalent Form to match with.
Important Words
at least
at most
must exceed
must not exceed
less than
more than
between
more than
Not less than
Less than
Not more than
© Project Maths Development Team 2013
Sample Sentence
Brian is at least 40
years old.
At most 40 people
attended a function.
Equivalent Form
Translation
Brian's age is greater
x ≥ 40, x ∈ R
than or equal to 40.
40 or fewer people
x ≤ 40, x ∈ N ∪ {0}
attended the
function.
Earnings must exceed Earnings must be
x > 40, x ∈ Q
€40.
greater than €40 per
hour.
The speed must not The speed is not
x ≤ 40, x ∈ R+
greater than 40km
exceed 40km per
per hour.
hour.
Spot's weight is less Spot’s weight is
x < 40, x ∈ R+
than 40kg.
below 40kg.
Dublin is more than is greater than
x>40,x ∈ R
40 miles away.
The film is between
The film is greater
40 ≤ x ≤ 50, x ∈ R
40 and 50 minutes
than or equal to
long.
40 minutes or less
than or equal to 50
minutes.
Mary spent €40 or
x ≥ 40, x ∈ Q
more.
Galway is not less
Galway is more than
x ≥ 40, x ∈ R
than 40km away.
or equal to 40km.
Damien is paid less
x < 40, x ∈ R+
than €40 per day.
There are some
There are less than or
x ≤ 40, x ∈ N
animals on the farm equal to 40 animals
but not more than
on the farm, given
40.
that there are some
animals on the farm.
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58
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