Transcript Document

Year 9 Inequalities
Dr J Frost ([email protected])
Objectives: Solving linear inequalities, combining inequalities and
representing solutions on number lines.
Last modified: 23rd March 2015
Writing inequalities and drawing number lines
You need to be able to sketch equalities and strict inequalities on a number line.
This is known as a
β€˜strict’ inequality.
x>3
Means: x is (strictly) greater
? than 3.
0
1
2
3
4
x < -1
Means: x is (strictly) less?than -1.
5
-3
-2
-1
?
4
5
?
2
x≀5
Means: x is greater than?or equal to 4.
3
1
?
xβ‰₯4
2
0
6
7
Means: x is less than or ?
equal to 4.
2
3
4
5
?
6
7
Deal or No Deal?
We can manipulate inequalities in various ways, but which of these are allowed and
not allowed?
𝒙>πŸ‘
Can we add
or subtract to
both sides?
π’™βˆ’πŸ>𝟐
Click to
οƒΌ Deal
Click to 
No Deal
Deal or No Deal?
We can manipulate inequalities in various ways, but which of these are allowed and
not allowed?
πŸπ’™ > πŸ”
𝒙>πŸ‘
Click to
οƒΌ Deal
Can we divide
both sides by
a positive
number?
Click to 
No Deal
Deal or No Deal?
We can manipulate inequalities in various ways, but which of these are allowed and
not allowed?
𝒙<𝟏
Can we multiply
both sides by a
positive number?
πŸ’π’™ < πŸ’
Click to
οƒΌ Deal
Click to 
No Deal
Deal or No Deal?
We can manipulate inequalities in various ways, but which of these are allowed and
not allowed?
𝒙<𝟏
Can we multiply
both sides by a
negative number?
βˆ’π’™ < βˆ’πŸ
Click to
Deal
Click to
No Deal
β€˜Flipping’ the inequality
If we multiply or divide both sides of the inequality by a
negative number, the inequality β€˜flips’!
OMG magic!
-2
2 < -4
4
Click to start
Bro-manimation
Alternative Approach
Or you could simply avoid dividing by a negative number at all by
moving the variable to the side that is positive.
βˆ’π‘₯ < 3
? π‘₯
βˆ’3 <
π‘₯ > ?βˆ’3
1 βˆ’ 3π‘₯ β‰₯ 7
1 βˆ’ 7 ?β‰₯ 3π‘₯
βˆ’6 β‰₯ ?3π‘₯
βˆ’2 β‰₯ ?π‘₯
?
π‘₯ ≀ βˆ’2
Quickfire Examples
2π‘₯ < 4
Solve
π‘₯ <? 2
βˆ’π‘₯ > βˆ’3
Solve
π‘₯ <? 3
4π‘₯ β‰₯ 12
Solve
π‘₯ β‰₯? 3
βˆ’4π‘₯ > 4
π‘₯
βˆ’ ≀1
2
Solve
π‘₯ <?βˆ’1
Solve
π‘₯ β‰₯?βˆ’2
Deal or No Deal?
We can manipulate inequalities in various ways, but which of these are allowed and
not allowed?
1
<2
π‘₯
Can we multiply
both sides by a
variable?
1 < 2π‘₯
Click to
Deal
Click to
No Deal
The problem is, we don’t know if the variable
has a positive or negative value, so negative
solutions would flip it and positive ones
wouldn’t. You won’t have to solve questions
like this until Further Maths A Level!
More Examples
3π‘₯ βˆ’ 4 < 20
Hint: Do the addition/subtraction before you do the
multiplication/division.
Solve
π‘₯<
? 8
4π‘₯ + 7 > 35
Solve
π‘₯ >? 7
π‘₯
5 + β‰₯ βˆ’2
2
Solve
π‘₯ β‰₯ ?βˆ’14
7 βˆ’ 3π‘₯ > 4
Solve
π‘₯
6βˆ’ ≀1
3
Solve
π‘₯ <? 1
π‘₯ β‰₯? 15
Dealing with multiple Hint:
inequalities
Do the addition/subtraction before you do the
multiplication/division.
8 < 5x
5x
-- 22 ≀ 23
and
2 < x and x ≀ 5
𝟐<π’™β‰€πŸ“
Click to start
bromanimation
More Examples
𝟏 < πŸπ’™ + πŸ‘ < πŸ“
βˆ’πŸ < βˆ’π’™ < πŸ’
Hint: Do the addition/subtraction before you do the
multiplication/division.
Solve
Solve
βˆ’πŸ < ?𝒙 < 𝟏
βˆ’πŸ’ < ?𝒙 < 𝟐
Test Your Understanding
𝟏𝟏 < πŸ‘π’™ βˆ’ πŸ’ < πŸπŸ•
𝟏 < 𝟏 βˆ’ πŸπ’™ < πŸ“
Solve
πŸ“ < 𝒙? < πŸ•
Solve
βˆ’πŸ < ?𝒙 < 𝟎
Exercise 1
Solve the following inequalities, and
illustrate each on a number line:
1
2
3
4
5
6
7
8
9
10
11
N1
2π‘₯ βˆ’ 1 > 5
𝒙 >?πŸ‘
βˆ’2π‘₯ < 4
𝒙 >?𝟐
5π‘₯ βˆ’ 2 ≀ 3π‘₯ + 4
𝒙 ≀?πŸ‘
N2
π‘₯
+1β‰₯6
𝒙 β‰₯?𝟐𝟎
4
𝑦
βˆ’1≀7
π’š ≀?πŸ’πŸ–
6
1βˆ’π‘¦
𝟏
≀𝑦
π’š β‰₯?
2
πŸ‘
1 βˆ’ 4π‘₯ > 5
𝒙 <?βˆ’πŸ
5 ≀ 2π‘₯ βˆ’ 1 < 9
πŸ‘ ≀ ?𝒙 < πŸ“
5 ≀ 1 βˆ’ 2π‘₯ < 9
βˆ’ πŸ’ < ?𝒙 ≀ βˆ’πŸ
10 + π‘₯ < 4π‘₯ + 1 < 33 πŸ‘ < 𝒙? < πŸ–
1 βˆ’ 3π‘₯ < 2 βˆ’ 2π‘₯ < 3 βˆ’ π‘₯ 𝒙 >?βˆ’πŸ
Sketch the graphs for
1
𝑦 = π‘₯ and 𝑦 = 1.
1
Hence solve π‘₯ > 1
0<x<1
?
You can get around the problem
of multiplying/dividing both sides
by an expression involving a
variable, by separately
considering when it’s positive,
and when it’s negative, and
putting this together.
Hence solve:
3
>4
π‘₯+2
If we assume 𝒙 + 𝟐 is positive, then 𝒙 >
πŸ“
βˆ’ 𝟐 and solving gives 𝒙 < βˆ’ . Thus βˆ’πŸ <
πŸ’
πŸ“
𝒙 < βˆ’ as we had to assume 𝒙 > βˆ’πŸ. If
πŸ’
?
πŸ“
𝒙 < βˆ’πŸ then this solves to 𝒙 > βˆ’ which
πŸ’
is a contradiction.
Thus βˆ’πŸ < 𝒙 < βˆ’
πŸ“
πŸ’
Combining inequalities
It’s absolutely crucial that you distinguish between the words β€˜and’ and β€˜or’ when
constraining the values of a variable.
AND
How would we express
β€œx is greater than or equal
to 2, and less than 4”?
? x<4
x β‰₯ 2 and
x β‰₯ 2,?x < 4
2 ≀ x? < 4
This last one emphasises the fact
that x is between 2 and 4.
OR
How would we express
β€œx is less than -1, or
greater than 3”?
? x>3
x < -1 or
This is the only way you would
write this – you must use the
word β€˜or’.
Combining inequalities
It’s absolutely crucial that you distinguish between the words β€˜and’ and β€˜or’ when
constraining the values of a variable.
2≀x<4
0
1
2
3
?
x < -1 or x > 4
4
5
-1
0
1
2
?
3
4
Combining inequalities
It’s absolutely crucial that you distinguish between the words β€˜and’ and β€˜or’ when
constraining the values of a variable.
To illustrate the difference, what happens when we switch them?
or
and
x β‰₯ 2 and x < 4
0
1
2
3
?
4
x < -1 or x > 4
5
-1
0
1
2
?
3
4
I will shoot you if I see any of these…
4>π‘₯<8
This is technically equivalent to:
x<4
?
4<π‘₯>7
This is technically equivalent to:
x>7
?
7>π‘₯>4
The least offensive of the three,
but should be written:
4<x<7
?
Combining Inequalities
In general, we can combine inequalities either by common sense, or using number lines...
2
5
Where are you on
both lines?
4
Combined
?
2
2<π‘₯<5
𝒙>πŸ“
5
4
π‘₯<4
Combined
?𝟐 < 𝒙 < πŸ’
Test Your Understanding
?
1st
2nd
-1
condition
condition
Combined
-3
?
3
?
?
5
Exercise 2
By sketching the number lines or otherwise,
combine the following inequalities.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?