# Order of operations Equations Formulae Removing brackets Factorising Inequations.

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Order of operations Equations Formulae Removing brackets Factorising Inequations

Hands up! 3 + 5 x 2 ? 13

A bit of order please! Brackets Of Divide Multiply Add Subtract B O D M A S ( 3 + 4 ) x 3= 7 x 3= 21 ( 3 + 4 ) x 3 5 + ½ of 8= 5 + 4= 9 5 + ½ of 8 3 + 5 x 2= 3 + 10= 13 3 + 5 x 2 18 - 12 ÷ 3= 18 - 4= 14 18 - 12 ÷ 3 These are the very last things you do! BOMDAS

BOMDAS Practice (1) 3 + 7 x 6 (2) (3 + 7) x 6 (3) 32 – 24 ÷ 4 (4) 5 + 6 x 3 - 1 (5) 5 + 6 x (3 – 1) = 3 + 42 = 45 = 10 x 6 = 60 = 32 - 6 = 26 = 5 + 18 – 1 = 22 = 5 + 6 x 2 = 5 + 12 = 17

Algebra Short hand 3a means 3 times a bc means b times c m/k means m divided by k x² means x times x efef means e divided by f

Formulae (1) Given that C = s + nr, calculate C when s=12, n=5 and r=8. C = s + nr BOMDAS C = 12 + 40 = 52 C = 12 +5x 85 x 8 M snr

p = 45 + 3 = 48 p = 5 x 9 + calculate p when a=5, s=9, d=18 and m=6. Given that p = as + dmdm p = as + dmdm 18 6 18 6 BOMDAS M D Formulae (2) as dmdm

s = 48 - 24 + 5 = 29 s = 3 x 4 x 4 - Given that s = 3t² - 6t + 5 calculate s when t=4. 6 x 4 BOMDAS M Formulae (3) s = 3t² - 6t + 5 + 5 3 x 4 x 4 6 x 4 3t² 6t

Now do Exercise 1 (HSDU Support Materials))

Removing brackets (1) 3(b + 5) = 3b 3 b This means 3 times b

Removing brackets (1) 3(b + 5) = 3b + 15 3 5 + This means 3 times 5

Removing brackets (2) 5(8 - f) = 40 5 8

Removing brackets (2) 5(8 - f) = 40 - 5f 5 f - Tip: Draw lines to keep you right. 5(8 – f) = 40 - 5f

Now do Exercise 2 Question 1 (HSDU Support Materials)) Algebra Menu

Removing brackets (3) 4(3b + 2) = 12b 4 3b 4 x 3b= 4 x 3 x b= 12 x b= 12b

Removing brackets (3) 4(3b + 2) = 12b + 8 4 2 +

Removing brackets (4) 3(7a – 2b) = 21a 3 7a

Removing brackets (4) 3(7a – 2b) = 21a - 6b 3 2b -

Removing brackets (5) c(3c + b) = 3c² c 3c c x 3c= 3c x c= 3c²

Removing brackets (5) c(3c + b) = 3c² + bc c b + Tip: Remember to draw lines to keep you right. c(3c + b) = 3c² + bc

Removing brackets (6) 6(5e + 3f - 2) = 30e 6 5e

Removing brackets (6) 6(5e + 3f - 2) = 30e 6 3f + 18f

Removing brackets (6) 6(5e + 3f - 2) = 30e 6 2 + 18f- 12 Careful! Tip: Remember to draw lines to keep you right. 6(5e + 3f -2) = 30e + 18f- 12

Now do Exercise 2 Questions 2 & 3 (HSDU Support Materials)) Algebra Menu

Simplifying (1) 4(3a + 5) - 8 = 12a 4 3a

Simplifying (1) 4(3a + 5) - 8 = 12a 4 5 + 20 This is not inside the brackets So don’t multiply! - 8- 8

Simplifying (1) 4(3a + 5) - 8 = 12a + 20 - 8- 8 This can be tidied up = 12a + 12

Simplifying (2) 3b + 2(5b – 6) = 10b 2 5b 3b + This is not inside the brackets So don’t multiply!

Simplifying (2) 3b + 2(5b – 6) = 10b 2 6 3b + - 12 Careful!

Simplifying (2) 3b + 2(5b – 6) = 10b3b + - 12 This can be tidied up = 13b - 12

Now do Exercise 2 Questions 4, 5. (HSDU Support Materials)) Algebra Menu

Factorising (1) 4a – 6b = 4 ? What’s the highest number that will divide into both 4 and 6? 6 This is called the highest common factor 2

? Factorising (1) 4a – 6b = 4a 2 2a ( ) 2

? Factorising (1) 4a – 6b = 6b 2 2a ( ) 3b - 2

Factorising (1) 4a – 6b = 2 2a ( ) 3b -

Factorising (2) 3b + bc = b ? What’s the highest common factor? b b This time it’s the letter b that they have in common.

? Factorising (2) 3b + bc = 3b b 3 b ( )

? Factorising (2) 3b + bc = bc b 3 c + b ( )

? Factorising (2) 3b + bc = b 3 c + ( )

Factorising (3) 6f + 9gf = ? What’s the highest common factor? 3 3 divides into 6 and 9. 6 9 f Notice that the letter f is also common to both. 9gf 6f6f

Factorising (3) 6f + 9gf = 3 f ( ) ? 2 6f

Factorising (3) 6f + 9gf = 3 f ( ) 2 9gf ? 3g +

Factorising (3) 6f + 9gf = 3 f ( ) 2 3g +

Now do Exercise 3 (HSDU Support Materials)) Algebra Menu

x + 5 = 20

x+5 20 (-5) x + 5 = 20

x 20 (-5) x + 5 = 20

x 15 x = 15 x + 5 = 20

(-5) (-5) x = 15 Example 1: x + 5 = 20

1)x + 8 = 122) a + 12 = 30 3)5 + x = 134)b + 7 = 42 (-8) x = 4 (-12) a = 18 (-5) x = 8 (-7) (-7) b = 35 Answers Exercise A

3x + 7 = 25

3x+7 25 (-7) 3x + 7 = 25

3x 25 3x + 7 = 25 (-7)

3x 18 3x = 18 3x + 7 = 25 ?

This means 3 times x equals 18 3 times 6 equals 18 So x must be 6 x = 6 3x = 18

3x + 7 = 25 (-7) (-7) 3x = 18 Example 2: x = 6

1) 2x + 7 = 23 2) 6a + 11 = 29 3) 5 +8y = 37 4) 9 + 3d = 42 (-7) (-7) 2x = 16 x = 8 (-11) (-11) 6a = 18 a = 3 (-5) (-5) 8y = 32 y = 4 (-9) (-9) 3d = 33 d = 11 Exercise B Answers

5x - 6 = 44

5x-6 44 (+6) 5x - 6 = 44

5x 44 5x - 6 = 44 (+6)

5x 50 5x = 50 5x - 6 = 44

(+6) (+6) 5x = 50 Example 3: x = 10

1) 5b - 8 = 122) 7a - 9 = 33 3) 6y - 6 = 304) 9z- 7 = 20 (+8) (+8) 5b = 20 b = 4 (+9) (+9) 7a = 42 a = 6 (+6) (+6) 6y = 36 y = 6 (+7) (+7) 9z = 27 z = 3 Exercise C Answers

Is there a shortcut?

5x - 6 = 44 Change side, change sign - 6

5x = 44 + 6 5x = 50 x = 10 Change side, change sign

5x - 6 = 44 5x = 44 + 6 5x = 50 Example 4: x = 10

3x + 7 = 25 Change side, change sign

3x = 25 - 7 3x = 18 x = 6 Change side, change sign

3x + 7 = 25 3x = 25 - 7 3x = 18 Example 5: x = 6

1) 2a + 7 = 23 2) 7x - 9 = 33 2a = 23 - 7 2a = 16 a = 8 7x = 33 + 9 7x = 42 x = 6 Exercise D Answers

6x = 9 + 2x Tricky one! 6x – 2x = 9 4x = 9 + 2x ?

4x = 9 Change side, change sign The 4 is multiplying the x, so when it changes side, it divides. 4

x = 9 ÷ 4 x = 2.25 4x = 9 Change side, change sign 4

5x + 2 = 3x + 6

5x+2 3x+6 (-2) 5x + 2 = 3x + 6

5x (-2) 3x+6

5x 3x+4 (-3x) 5x + 2 = 3x + 6

5x 4 (-3x)

2x 4 2x = 4 5x + 2 = 3x + 6 So x = 2

Change side, change sign 5x = 3x + 6 - 2 5x = 3x + 43x 5x –3x = 4 2x = 4 5x + 2 = 3x + 6 x = 2 + 2

Change side, change sign 3x + 6 + 4 = 8x 3x + 10 = 8x3x 10 = 8x – 3x 10 = 5x 3x + 6 = 8x - 4 x = 2 - 4 Most Xs are on the right. So this time: Xs on the right, numbers on the left! 3x 8x

3x + 6 = 8x - 4 6+4 = 8x – 3x Example 6: 10 = 5x x = 2 Try two changes together!

1) 5a + 7 = 2a + 22 2) 3x + 9 = 7x - 15 5a – 2a = 22 - 7 3a = 15 a = 5 9 + 15 = 7x – 3x 24 = 4x x = 6 Exercise E Answers

is greater than 9 8 Inequations 9 8 >

is less than Inequations 5 9 < 5 9

Hint: The less than sign is almost L shaped. Remember is greater than is less than < ess than < <

Practice: fill in the correct sign (1) 7 10(2) 12 9 (3) 2 3(4) 5 0 (5) 3 -2(6) -6 -3 (7) -2 2(8) -1 0 (9) -4 -5(10) 3 3 < > < > > < < < > < Click here for number line

To drive, what age do you have to be? To get married in Scotland, what age do you have to be? To buy alcohol, what age do you have to be? Click on the picture to reveal the answer.

What speed should you drive at in town? Click on the picture to reveal the answer. How many passengers can a black taxi take?

is greater than or equal to ABCD Next question

is less than ABCD

is greater than ABCD

is less than or equal to ABCD Finish

3a + 8 < 26 Solving inequations + 8 3a < 18 a < 6 - 8 Use the same method as for equations!

3a + 8 < 26 3a < 26 - 8 3a < 18 Example: a < 6

Now do Exercise ? (HSDU Support Materials))

K Hughes 2003

Aged greater than or equal to 18

Aged greater than or equal to 17

Aged greater than or equal to 16

Speed less than or equal to 30 mph

Passengers less than or equal to 5

Number Line 01234567-2-3-4-5-6-7 positive numbersnegative numbers Negative Positive