yr 9 equations_ and inequalities teacher

Equa

tion

s&

Ineq

uali

ties

& INEQUALITIESEQUATIONS

SOLUTIONS

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1100% Equations & Inequalities Solutions

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Equations & Inequalities

SERIES TOPIC

J 6

Solutions Basics

a x + 3 = 6 is a linear / quadratic equation

(the highest power of x is 1 so this is a linear equation)

b 3x2 = 12 is a linear / quadratic equation

(the highest power of x is not 1 (it’s 2) so this is not a linear equation)

c 3x = 12 is a linear / quadratic equation


d x25 = 3 is a linear / quadratic equation


2. Solve these linear equations:

a 4 7

4 4 7 4

x

x x

x 3

+ =

+ = +

=

b 3 10

3 3 10 3

x

x

x 7

+ =

+ =

=

c 2 13

2 2 13 2

x

x

x 15

=

+ = +

=

d 6 10

6 6 10 6

x

x

x 16

=

+ = +

=

Page 4 questions

1. Choose whether the following equations are linear or quadratic by circling the appropriate word (don’t solve):

Subtract 4 from both sides


Add 6 to both sides

Add 2 to both sides

2 100% Equations & Inequalities Solutions



SERIES TOPIC

J 6

BasicsSolutions

Page 5 questions

3. Find the value of the variable in each of these linear equations:

a x

x

x

2 8

22

28

4

=

=

=

c x

x

x

45

44 5 4

20

# #

=

=

=

e x

x

x

x

x

23 9

23 2 9 2

3 18

33

318

6

# #

=

=

=

=

=

b x

x

x

5 35

55

535

7

=

=

=

d x

x

x

38

33 8 3

24

# #

=

=

=

f x

x

x

x

x

35 5

35 3 5 3

5 15

55

515

3

# #

=

=

=

=

=

Divide both sides by 2

Multiply both sides by 4

Multiply both sides by 2 Multiply both sides by 3

Divide both sides by 3 Divide both sides by 5






SERIES TOPIC

J 6

Solutions Basics

a

4 20

x

x

x

x

x

4 3 23

4 3 23 3

4 20

4 4

5

+ =

=

=

=

=

c m

m

m

m

m

5 6 31

5 6 6 31 6

5 25

55

525

5

+ =

+ =

=

=

=

e k

k

k

2 10

22

210

5

=

=

=

b a

a

a

a

a

2 5 9

2 5 5 9 5

2 14

22

214

7

=

+ = +

=

=

=

d n

n

n

n

n

4 7 24

4 4 7 24 4

7 28

77

728

4

+ =

+ + = +

=

=

=

f Hint( : )m mm

m

m

m

m

14 6

4 4 6 4

2

1

1

12

2

=+ =

+ =

=

=

=

Page 6 questions

4. Solve for the variable in each linear equation:


Subtract 6 from both sides Add 4 to both sides


Add 5 to both sides



Divide both sides by –2


Divide both sides by –1





SERIES TOPIC

J 6

BasicsSolutions

a 4

4 4 4

y

y

y

4

4

16

# #

=

=

=

c

2 54

p

p

p

p

p

9

26

9

29 6 9

2 54

2 2

27

# #

=

=

=

=

=

e 14

3 14 3

7 42

x

x

x

x

x

37

37

77

742

6

# #

=

=

=

=

=

b 15

4 15 4

5 60

x

x

x

x

x

45

45

55

560

12

# #

=

=

=

=

=

d

4 88

d

d

d

d

d

114 8

114 11 8 11

4 88

4 4

22

# #

=

=

=

=

=

f

10 100

m

m

m

m

m

410 25

410 4 25 4

10 100

10 10

10

# #

=

=

=

=

=

a

so or

y

y

y

y

y

y y

9 0

9 9 0 9

9

3

3

3 3

2

2

2

2 2!

!

=

+ = +

=

=

=

= =

^ h

b

so or

4 100

25

5

5 5

x

x

x

x

x

x x

44

4100

5

2

2

2

2 2!

!

=

=

=

=

=

= =

^ h

Page 7 questions

5. Solve for the variable in each linear equation:

6. Solve these quadratic equations:




Add 9 to both sides

Since –32 = 9 and 32 = 9

Find the square root of both sides








Find the square root of both sides






SERIES TOPIC

J 6

Solutions Knowing More

a

1

1

110

u u

u u u u

uu

u

2 10 3

2 3 10 3 3

1 10 10 0 10

10

=

=

+ = +

=

=

b 7 18 3 10

7 3 18 3 3 10

4 18 18 10 18

4 28

x x

x x x x

x

xx

x44

428

7

= +

= +

+ = +

=

=

=

c x

x

x

x

x

x

3 2 3

3 6 3

3 6 6 3 6

3 9

33

3

9

3

+ =

+ =

+ =

=

=

=

^ h

d 4 18 12 2

4 2 18 12 2 2

6 18 18 12 18

6 6

y y

y y y y

y

y

y 1

+ =

+ + = +

+ =

=

=

e 10 2

10 60 20 2

10 2 60 20 2 2

8 60 20

8 60 60 20 60

8 80

n n

n n

n n n n

n

n

nn

n

6 10

88

880

10

= +

= +

= +

=

+ = +

=

=

=

^ ^h h

Page 11 questions

1. In these linear equations the variable appears on boths sides. Solve for the missing value:

Subtract 3u from both sides

Subtract 3x from both sides

Add 10 to both sides


Divide both sides by -1


Expand the brackets





Expand the brackets



Subtract 2n from both sides

Add 2y to both sides




SERIES TOPIC

J 6

Knowing MoreSolutions

f 6 4 5

6 4 5 15

6 5 4 5 5 15

11 4 15

11 4 4 15 4

11 11

m m

m m

m m m m

m

m

mm

m

3

1111

1111

1

= +

=

+ = +

=

+ = +

=

=

=

^ h

g 8 5 3 4

8 32 5 3 4

3 29 4

3 29 29 4 29

3 33

k k

k k

k

k

kk

k

4

33

333

11

+ =

+ =

=

+ = +

=

=

=

^ h

h 5 6 3 6

10 5 6 12 3 6

4 10 6

4 10 10 6 10

4 4

y y

y y

y

y

yy

y

2 1 2

44

44

1

+ =

+ + =

+ =

+ =

=

=

=

^ ^h h

Page 11 questions


Expand the brackets

Add 5m to both sides

Add 4 to both sides


Expand the brackets



Collect like terms

Expand the brackets



Collect like terms




SERIES TOPIC

J 6


i 8 12

8 2 18 12

6 18 12

6 18 18 12 18

6 30

t t

t t

t

t

t

t

t

2 18

66

6

30

5

=

+ =

+ =

+ =

=

=

=

^ h

j 2 3 10

2 6 3 12 10

6 10

6 6 10 6

4

a a

a a

a

a

aa

a

3 4

1 1

4

4

+ + =

+ =

=

+ = +

=

=

=

^ ^h h

Page 11 questions


2. Find Ivan's mistake when he tried to solve this equation?

3 2 2 72 2 2 2

h h

h h

h h

h h

h

h h

3 2 2 1 5

3 2 2 2 5

3 2 2 7

5

+ = + +

+ = + +

+ = +

+ = +

=

^ ^h h



Expand the brackets

Expand the brackets

Oops - Expanded the brackets incorrectly

should be 3h + 6 = 2h + 2 + 5

Correct answer then comes out to h = 1

Add 6 to both sides

Divide both sides by -1

Collect like terms




SERIES TOPIC

J 6


a 1 4

1 1 4 1

5

8 5 8

x

x

x

x

x

8

8

8

840

# #

=

+ = +

=

=

=

b n n

n n

n n

n n

n n n n

n

24

125 5

122

4 12125 5

212 48

1260 60

6 48 5 60

6 5 48 48 5 5 60 48

12

# #

+ = +

+ = +

+ = +

+ = +

+ = +

=

` `j j

c b

b

b

b

b

b

53 4 4

3 4 5 4

3 4 20

3 4 4 20 4

3 24

8

#

+ =

+ =

+ =

+ =

=

=

Add 1 to both sides


Multiply both sides by denominator 5

Lowerst common Denominator (LCD) is 12

Multiply both sides by (LCD)

Expand brackets

Page 12 questions

3. Solve these linear equations which contain fractions:




SERIES TOPIC

J 6


d c c

c c

c c

c

c

c

212

22

2 12

2 24

3 24

33

324

8

# #

+ =

+ =

+ =

=

=

=

` j

e r

r

r

r

r

r

r

516 2 10

16 2 10 5

16 2 50

16 2 2 50 2

16 48

1616

1648

3

#

+ =

+ =

+ =

+ =

=

=

=

f m m

m m

m m

m m

m

m

m

2 35

62 3

6 5

26

36 30

3 2 30

5 30

55

530

6

# #

+ =

+ =

+ =

+ =

=

=

=

` j

Page 12 questions


Multiply both sides by LCD (2)


Multiply both sides by LCD







SERIES TOPIC

J 6


a q q

q q

q q

q q

q q q

q

q

q

3

7 5

6

4 30

63

7 56

6

4 30

2 7 5 4 30

14 10 4 30

14 4 10 10 4 30 10

10 40

1040

4

# #

#

+=

+=

+ =

+ =

+ =

=

=

=

c c

^

m m

h

b

u u

u u

u u

u u

u

u

u

u

u

65 5

95 10 2 7

186

5 5 189

5 10 18 2 18 7

3 5 5 2 5 10 36 126

15 15 10 20 36 126

25 71 126

25 71 71 126 71

25 55

2525

2555

511

# # # #

# #

+ + + + =

+ + + + =

+ + + + =

+ + + + =

+ =

+ =

=

=

=

` `

^ ^

j j

h h

c

g g g

g g g

g g g

g g g

g g g g

g

3

2

10 21

4

3

603

260

1060

21 60

4

3

20 2 6 30 15 3

40 6 30 45

46 45 30 45 45

30

# # # #

# #

+ = +

+ = +

+ = +

+ = +

= +

=

Page 13 questions


LCD is 6

Multiply by LCD

LCD is 60

Multiply by (LCD)

Multiply by LCD

LCD is 18




SERIES TOPIC

J 6


d x

xx

x

x

x

x

2 6

2 6

2 6

22

26

3

# #

=

=

=

=

=

e

d

dd

d

d

d

d

d

d

8 26 1

8 28 2

6 1 8 2

6 8 2

6 2 8 2 2

8 8

88

88

1

# #

=

=

=

+ = +

=

=

=

^ ^h h

f 2

2

3 45 2

3 2 45 2 2 45

k

k

kk

kk

k k

k k k k

k

3 45

3 45

45

# #

=

=

=

+ = +

=

questions

Multiply both sides by the denominator


You could rewrite this as 8d - 2 = 6 if you prefer





SERIES TOPIC

J 6


questions

5. Three times a number is 45. What is the number?

Let the number be n

3 45nn

n33

345

15

=

=

=

6. Claire, Leanne and Lindsay are sisters. Claire is two years older than Leanne and Leanne is 4 years older than Lindsay. The sum of all their ages is 54. How old is each sister?

Let Leanne's age be x

Ages: Claire + Leanne + Lindsay = 54

54

3 2 54

3 2 2 54 2

3 56

x x x

x

x

xx

x

2 4

33

356

1832

+ + + =

=

+ = +

=

=

=

^ ^h h

So Leanne is 1832 , Claire is 20

32 and Lindsay is 14

32

7. Charlie has been collecting stamps which he keeps in two separate books. The second book has 7 more than triple the stamps of the first book. If he has 35 stamps in total (from both books) then:

Let the number of stamps in the first book be xThe number of stamps in the second book is 3x + 7

Book book total stamps1 2

3 7 35

4 7 35

4 7 7 35

4 7 7 35 7

4 28

x x

x

x

x

xx

x44

428

7

+ =

+ + =

+ =

+ =

+ =

=

=

=

a How many stamps are in the first book? 7

b How many stamps are in the second book? 35 - 7 = 28




SERIES TOPIC

J 6


Page 14 questions

8. Victor has a bag filled with 2c and 5c coins. He has 2 more coins worth 2c than the coins worth 5c. How many 2c coins does Victor have if all his coins sum to 88c?

Let n be the number of 5c coins. He has (n + 2) of 2c coins.

They all add up to 88c so,

2 5 88

2 4 5 88

7 4 88

7 4 4 88 4

7 84

n n

n n

n

n

nn

n

2

77

784

12

# #+ + =

+ + =

+ =

+ =

=

=

=

^ h

So he has 12 coins worth of 5c and 14 coins worth of 2c.




SERIES TOPIC

J 6

Using Our KnowledgeSolutions

2. Solve these inequalities:

a

c

b

d

3 4

3 3 4

x

x

x 1

1

1

1

+

+

4 5

4 4 5 4

x

x

x 9

$

$

$

+ +

m

m

m

7 4

7 7 4 7

11

$

$

$

+

+

10 8

10 10 8 10

p

p

p 2

#

#

#

+ +

Page 18 questions

1. Identify if the following are true or false:

a 6 32 True

b 5 81 True

c 3 82 False

d 2 51 False

e 4 42 False

f 8 41 True




SERIES TOPIC

J 6

Solutions Using Our Knowledge

Page 18 questions

2. Solve these inequalities:

e

g

i

f

h

j

q

q

q

5 35

5

5

535

7

#

#

#

12

h

h

h

h

h

4 3 51

4 3 3 51 3

4 48

44

448

2

2

2

2

2

+

+

x x

x x x x

x

x

x

5 24

5 24

4 24

44

424

6

2

2

2

2

2

+

+

x x

x x

x x x x

x

x

3 1 3 1

3 3 3 3

3 3 3 3 3 3 3 3

6 6

1

#

#

#

#

#

+ + + +

^ ^h h

h

h

h

h

h

h

h

2

4 82

22

4 82 2

4 8 4

4 8 8 4 8

4 4

44

44

1

# #

1

1

1

1

1

1

1

+ +

c m

14

20 20 14 20

5 3 4 2 280

15 8 280

7 280

x x

x x

x x

x x

x

x

x

43

52

43

52

77

7280

40

# # #

# #

$

$

$

$

$

$

$




SERIES TOPIC

J 6


Page 19 questions

3. Graph these inequalities:

a x 32

c x 0#

e x5 51#

g and0 3x x1 $

b

x 2$

d x1 31 1

f x2 41 #

h and4 1x x2 #

-5 -4 -3 -2 -1 0 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5

4. Write down the inequality represented by each of the following graphs:

-5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5

a b

c d

e f

x 41 4x #

1x $- x2 41#-

x4 01 #- 3x #-




SERIES TOPIC

J 6

Solutions Using Our Knowledge

Page 20 questions

5. Solve these more complicated linear inequalities, then graph their solution:

a 5 6 9

5 6 9 18

5 9 6 6 18 6

4 12

x x

x x

x x

xx

x

2

4

4

412

3

2

2

2

2

1

1

+ +

+ +

+

^ h

Reverse the inequality when dividing by a negative number

c 2

8 8 2

4 3 16

4 4 3 16 4

3 12

4

d

d

d

d

d

d

d

d

8

4 3

8

4 3

3

3

312

4

# #

#

#

#

#

#

#

#

$

c m

Reverse the inequality when multiplying by a negative number

b 7 4 5

7 4 16 5

3 16 16 5 16

3 21

y y

y y

y

y

y

y

4

33

321

7

2

2

2

2

2

2

+

+ +

^ h

-2 -1 0 1 2 3 4 5 6 7 8-5 -4 -3 -2 -1 0 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4 5




SERIES TOPIC

J 6


questions

5. Solve these more complicated linear inequalities, then graph their solution:

d b b

b b

b b

b

b

b

b

8 35

248 3

5 24

3 8 120

5 120

55

5120

24

24

# #

#

#

#

#

#

#

$

` j

Multiply both sides by LCM of all fractions (24)

Reverse the inequality when multiplying by a negative number.

f

.

c c

c c

c c

c c c c

c

c

c

48

6 2

21

12

4848

6 248

21 48

12

6 2 24 4

6 4 2 2 24 4 4 2

10 26

1010

1026

2 6

# # #

1

1

1

1

1

1

1

+ + + +

c m


Reverse the inequality when multiplying by a negative number

e a a

a a

a a

a a a a

a

a

a

a

2 104 3

102

1010

4 3

5 4 3

5 4 4 4 3

9 3

99

93

31

31

# #

$

$

$

$

$

$

$

#

+

+

+

+

` `j j


Reverse the inequality when multiplying by a negative number22 23 24 25 26 27 28 29 30 31 32

-5 -4 -3 -2 -1 0 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4 5




SERIES TOPIC

J 6

Solutions Thinking More

Page 26 questions

1. Write down 2 possible solutions for the variables in these equations:

a x + y = 4

Anything where x and y add up to 4

Eg x = 3, y = 1 or x = 2, y = 2 or x = 4, y = 0 or x = 5, y = -1 or x = 3.5, y = 0.5 etc

Can also write these pairs like this: (3, 1), (2, 2) (4, 0), (5,-1), (3.5, 0.5)

b 2a + b = 6

Anything where twice a, plus b, makes 6.

Eg a = 1, b = 4, or a = 2, b = 2 or a = 3, b = 0 or a = 4, b = -2 or a = 0.5, b = 5 etc

c 3x - 4y = 10

Anything where 3 times the first number, minus three times the second number, makes 10.

Eg x = 6, y = 2, or x = -2, y = -4 or x = 30, y = 20 or x = 2, y = -1 or x = 1, y = -1.75 etc

2. Solve for the variables in these simultaneous equations using the substitution method:

a 2x + y = -1 1

x - 2y = -4 2

Make one of the variables the subject of 1

y = -1-2x

Substitute this into 2

2 4

2 4 4

5 2 4

5 6

.

x x

x x

x

x

x

1 2

56 1 2

=

+ + =

+ =

=

= =

^ h

Substitute this x value into either equation, say 2

2 4

1.2 2 4

2 4 1.2

.

.

x y

y

y

y

y2

2 8

1 4

=

=

= +

=

=

so x = -1.2 and y = 1.4

b 2p + 3q = 10 1

2q - 4p = 44 2

Make one of the variables the subject of 2 , say q

2 44 4q pq p

q p2

2

244

2

4

22 2

= +

= +

= +

Substitute this into 1

2 3 10

2 66 6 10

8 66 66 10 66

8 56

p p

p p

p

p

p

22 2

7

+ + =

+ + =

+ =

=

=

^ h

Substitute p back into either equation to solve for q q

q

q

q

q

2 7 3 10

14 3 10

3 14 14 10 14

3 24

8

# + =

+ =

+ = +

=

= so p = -7 and q = 8

1




SERIES TOPIC

J 6

Thinking MoreSolutions

Page 27 questions

3. Use the graphical method to solve for these equations:

-2 0 2 4

4

2

-2

-4

-6

y

x

987654321

-1-2

-7 -6 -5 -4 -3 -2 -1 0 1 2

y

x

a 3x + 2y = 2 1

2x - y = 6 2

Make y the subject of both equations

y x

y x

2

31

2 6

= +

=

The lines intersect at (2,-2) so x = 2 and y = -2

b 3y - 4x = 24 1

2y + 2x = 2 2

Make y the subject of both equations

y x

y x34 8

1

= +

= +

The lines intersect at (-3, 4) so x = -3 and y = 4

2

1




SERIES TOPIC

J 6


Page 28 questions

4. Solve for the variables in these simultaneous equations using the elimination method:

a 3x - y = -15 1

y + 2x = 0 2

In this case we can eliminate y by adding the two equations together.

x y

y x

x x

x

x

3 15

2 0

3 2 15

5 15

3

=

+ =

+ =

=

=

1

2

1 + 2

Substitute x = -3 into either equation – say 1 .

3 3 15

9 15

y

y

y 6

# =

=

=

so x = -3 and y = 6

b b - 4a = -12 1

3a - 2b = -1 2

Make the coefficient of b the same in both equations; then subtract 2 from 1

2b - 8a = -24 1 # 2 3a - 2b = -1 2

5 25a

a

a

55

525

5

- =-

-- =

--

=

(2 # 1 ) + 2

Substitute a = 5 into either equation, say 2

3 5 2 1

15 2 1

2 15 15 1 15

2 16

b

b

b

b

b 8

# =

=

+ =

=

=

So a = 5 and b = 8




SERIES TOPIC

J 6


a 7m = 16 - 6n

2n = 3m + 16

Page 29 questions

5. Solve these simultaneous equations using any method:

Page 30 questions


a 8c - 8d = 2

9c - 8d = 5

1

1

1

2

2

23c

c 3`

- =-

=

-

substitute c 3= into 1 to find d:

Substitute this into 1 :

Substitute this into 1 to solve for n:

d

d

d

8 3 8 2

8 22

822

411

`

`

`

- =

=

= =

^ h

and3c d411` = =

In 2 , make n the subject (find n by itself):

n m2

3 16= +

m m

m m

7 16 62

3 16

7 16 9 48

`

`

= - +

= - -

` j

2 3 16n

n

2

5

= - +

=

^ h

16 32

2

m

m

`

`

=-

=-

Now, solve for m:




SERIES TOPIC

J 6


Page 31 questions


a 6a - 2b - 10 = 0

2a + 3b - 29 = 0

1

1

2

2

Multiply 2 by 1 and subtract:

Substitute this into 2 :

6 2 10 0a b

a b6 9 87 0

- - =

+ - =

11 77

7

b

b`

- =-

=

2 3 2 0

2 8

a

a

a

7 9

4

+ - =

=

=

^ h




SERIES TOPIC

J 6


Page 32 questions

8. Find equations and solve them for these word problems (using any method):

a The sum of two numbers is 12. The sum of the first number, and double the second number is 16. What are the numbers?

Call the numbers a and b

a + b = 12 1 (The sum of two numbers is 12)

a + 2b = 16 2 (The sum of the first number, and double the second number is 16)

You can solve this using any of the 3 methods, for example elimination. We can choose to eliminate a since it already has the same coefficient in both equations.

b - 2b = 12 - 16 1 - 2

-b = -4

b = 4

Substitute b = 4 into 1

a + 4 = 12

a = 8

So a = 8 and b = 4

b Ari is three years older than Eric. In three years from now, Ari will be twice as old as Eric will be. How old are

they now?

Lets call Eric’s present age, e, and Ari’s present age a

a = e + 3 1

3a e23+ = + 2

You can solve this using any of the 3 methods, for example substitution.

Substitute 1 into 2

e e

e e

e e

e

23 3 3

226 2 3

6 2 6

0

# #

+ + = +

+ = +

+ = +

=

` ^j h

Substitute e = 0 into either equation gives a = 3

So Eric is 0; Ari is 3.




SERIES TOPIC

J 6


Page 32 questions

8. Find equations and solve them for these word problems (using any method):

c A resturaunt sells two kinds of meals: pizza and pasta. A pizza costs $14 and a pasta costs $10. In a single day the resturaunt sold 79 meals. If they earned $994 on this day, how many of each meal was sold?

If we call number of pizzas sold is x and number of pasta meals sold is y, then x+ y = 79 (79 meals sold in total)

14x + 10y = 994 ($14 for every pizza + $10 for every pasta meal = $994 in total) You can solve this with any of the 3 methods, for example graphically

Careful measurement will show that x = 51 and y = 28. So 51 pizzas and 28 pasta meals were sold.

Note: This method is approximate, and the answer will not always be exact.

y

x -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95

95

90

85

80

75

70

65

60

55

50

45

40

35

30

25

20

15

10

5

0-5

-10

-15

-20

-25

-30

-35




SERIES TOPIC

J 6

Notes

yr 9 equations_ and inequalities teacher

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