number and algebraequations and inequalitiesweb2.hunterspt-h.schools.nsw.edu.au/studentshared...6-07...

6Number and Algebra

EquationsandinequalitiesMany scientific, natural and social phenomena can bemodelled by equations and inequalities.Historically, algebra dates back to ancient Egypt andBabylon, where linear and quadratic equations were solved.In ancient Babylon, quadratic equations were solved by verysimilar methods to those still relevant and taught today.

n Chapter outlineProficiency strands

6-01 Equations U F R6-02 Equations with

algebraic fractions* U F R6-03 Quadratic equations

x2 þ bx þ c ¼ 0* U F R C6-04 Equation problems U F PS R C6-05 Equations and

formulas* U F PS R C6-06 Graphing

inequalities on anumber line* U F C

6-07 Solving inequalities* U F R

*STAGE 5.2

nWordbank> The symbol for ‘is greater than’

< The symbol for ‘is less than’

equation A mathematical statement that two quantities areequal, involving algebraic expressions and an equals sign(¼)

formula A rule written as an algebraic equation, usingvariables

inequality A mathematical statement that two quantitiesare not equal, involving algebraic expressions and aninequality sign (>, �, <, or �)

quadratic equation An equation involving a variablesquared (power of 2), such as 3x2 � 6 ¼ 69.

solution The answer to an equation, inequality orproblem; the correct value(s) of the variable that makes anequation or inequality true

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NEW CENTURY MATHSfor the A u s t r a l i a n C u r r i c u l u m 10

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n In this chapter you will:• (STAGE 5.2) solve linear equations involving simple algebraic fractions• (STAGE 5.2) solve simple quadratic equations using a range of strategies• (STAGE 5.2) substitute values into formulas to determine an unknown• (STAGE 5.2) solve linear inequalities and graph their solutions on a number line• solve linear equations and problems involving equations

SkillCheck

1 Solve each equation.

a 8y ¼ 16 b 10x ¼ 120 c m5¼ 2

d w þ 6 ¼ 10 e m � 3 ¼ 12 f n þ 6 ¼ �4

2 Expand each expression.

a 5(x þ 10) b 4(y � 1) c 2(5 � 3y)


a 2x þ 3 ¼ 23 b 3x � 5 ¼ 19 c 4a þ 5 ¼ 2a � 19

d 3xþ 25¼ 4 e 4(2 � x) ¼ �24

6-01 Equations

Example 1

Solve each equation.

a 3m � 6 ¼ 12 b 5 � 2a ¼ 3a

c 9x þ 10 ¼ 7x � 6 d 5(p þ 6) ¼ 3p þ 5

Solutiona 3m� 6 ¼ 12

3m� 6þ 6 ¼ 12þ 6

3m ¼ 183m

3¼ 18

3m ¼ 6

Check:LHS ¼ 3 3 6� 6 ¼ 12

RHS ¼ 12

LHS ¼ RHS

Adding 6 to both sides.

Dividing both sides by 3.

Worksheet

StartUp assignment 6

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Equations and inequalities

188

b 5� 2a ¼ 3a

5� 2aþ 2a ¼ 3aþ 2a

5 ¼ 5a

5a ¼ 55a

5¼ 5

5a ¼ 1

Check:LHS ¼ 5� 2 3 1 ¼ 3

RHS ¼ 3 3 1 ¼ 3

LHS ¼ RHS

Adding 2a to both sides.

Dividing both sides by 5.

c 9xþ 10 ¼ 7x� 6

9xþ 10� 7x ¼ 7x� 6� 7x

2xþ 10 ¼ �6

2xþ 10� 10 ¼ �6� 10

2x ¼ �162x

2¼ �16

2x ¼ �8

d 5ðpþ 6Þ ¼ 3pþ 5

5pþ 30 ¼ 3pþ 5

5pþ 30� 3p ¼ 3pþ 5� 3p

2pþ 30 ¼ 5

2pþ 30� 30 ¼ 5� 30

2p ¼ �252p

2¼ �25

2

p ¼ �1212

Example 2

Solve 4(y þ 1) þ 3(y � 5) ¼ 8

Solution4ðyþ 1Þ þ 3ðy� 5Þ ¼ 8

4yþ 4þ 3y� 15 ¼ 8

7y� 11 ¼ 8

7y� 11þ 11 ¼ 8þ 11

7y ¼ 197y7¼ 19

7

y ¼ 2 57

Check:

LHS ¼ 4ð2 57þ 1Þ þ 3ð2 5

7� 5Þ

¼ 8

RHS ¼ 8

LHS ¼ RHS.

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Exercise 6-01 Equations1 Solve each equation,

a k6¼ 10 b w þ 3 ¼ �6

c 5y � 1 ¼ 9 d 3a þ 10 ¼ 25e 2x þ 6 ¼ 22 f 15a � 2 ¼ 13g 12 � r ¼ 18 h 7w � 10 ¼ 32i 9y � 6 ¼ �24 j 11 � 6a ¼ �10k 9y ¼ 3y þ 32 l 5a ¼ a � 7

2 What is the solution to 6x � 3 ¼ 27? Select the correct answer A, B, C or D.

A x ¼ 4 B x ¼ 5 C x ¼ 10 D x ¼ 18

3 What is the solution to 10 � 2a ¼ 20 ? Select A, B, C or D.

A a ¼ �15 B a ¼ 8 C a ¼ 32 D a ¼ �5


a 5y þ 10 ¼ 3y þ 30 b 8a þ 20 ¼ 4a þ 10c 6y � 1 ¼ 3y þ 14 d 12a þ 30 ¼ 5a þ 9e 5y þ 3 ¼ 8y � 21 f 14x � 20 ¼ 8x � 14g 9y þ 1 ¼ 3y � 5 h 15x � 15 ¼ 8x � 85i 8m � 10 ¼ 5 � 2m j 18 � 3y ¼ 6 � 2y

k 1 � 7a ¼ 10 þ 2a l 11 � 5x ¼ 3x þ 43

5 Solve 4y ¼ y � 15. Select A, B, C or D.

A y ¼ �3 B y ¼ 74

C y ¼ �5 D y ¼ 11


a 3(x � 6) ¼ 30 b 5(m þ 10) ¼ 80c 2(5y þ 3) ¼ 46 d 3(y þ 2) ¼ 5y � 10e 5(y þ 4) ¼ 3y þ 6 f 10(x � 3) ¼ 5(x þ 5)g 2(3m þ 6) ¼ 4(m � 1) h 5(2a þ 7) ¼ 5(4 � a)i 3(1 � 2y) ¼ 18 � 3y

7 Solve 2(y � 3) ¼ 5 þ 4y. Select A, B, C or D.

A y ¼ �9 B y ¼ �5 C y ¼ � 112

D y ¼ � 12


a 3(d þ 3) þ 4(d þ 1) ¼ 15 b 3(y � 1) þ 5(y þ 4) ¼ 10c 7(k þ 1) þ 2(k � 6) ¼ 3 d 5(g � 3) þ 2(g � 2) ¼ 4e 6(2h þ 3) þ 5(h � 3) ¼ 9 f 2(1 þ p) þ 3(4 þ p) ¼ 5

See Example 1

Stage 5.2

See Example 2

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6-02 Equations with algebraic fractions

Example 3


a yþ 25¼ �1 b 3ðp� 3Þ

2þ 6 ¼ 1

Solutiona yþ 2

53 5 ¼ �1 3 5

yþ 2 ¼ �5

yþ 2� 2 ¼ �5� 2

y ¼ �7

b 3ðp� 3Þ2

þ 6 ¼ 1

3ðp� 3Þ2

þ 6� 6 ¼ 1� 6

3ðp� 3Þ2

¼ �5

3ðp� 3Þ2

3 2 ¼ �5 3 2

3ðp� 3Þ ¼ �10

3p� 9 ¼ �10

3p� 9þ 9 ¼ �10þ 9

3p ¼ �13p

3¼ �1

3

p ¼ � 13

Investigation: Make your own equation

Here are two equations that have the same solution, x ¼ 6:

5x � 1 ¼ 23 þ x and 3xþ 1210

¼ 3

1 For each solution below, make up two equations that have that solution.

a x ¼ 4 b x ¼ 12

c x ¼ 10

d x ¼ 1.5 e x ¼ 0 f x ¼ �22 Compare your answers with those of other students. Check that each equation is correct.

Stage 5.2

Puzzle sheet

Equations code puzzle

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Puzzle sheet

Equations order activity

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Puzzle sheet

Solving linearequations 1

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Puzzle sheet

Solving linearequations 2

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Example 4


a 2aþ 45¼ 2

3b 2m

3� m

2¼ 2

Solutiona 2aþ 4

5¼ 2

3For equations where all terms are fractions, multiply both sides by a common multiple of thedenominators to remove the fractions.The lowest common multiple (LCM) of 5 and 3 is 15, so multiply both sides by 15.2aþ 46 51

3 153 ¼ 26 31

3 155

3ð2aþ 4Þ ¼ 10

6aþ 12 ¼ 10

6a ¼ �2

a ¼ �26

a ¼ � 13

b 2m3� m

2¼ 2

Multiply both sides by 6, the LCM of 3 and 2.

62m

3� m

2

� �¼ 6 3 2

6 62 32m

6 31�6 63 3

m

6 21¼ 12

4m� 3m ¼ 12

m ¼ 12

Exercise 6-02 Equations with algebraic fractions1 What is the solution to each equation? Select the correct answer A, B, C or D.

a 3y4¼ 6

A y ¼ 9 B y ¼ 8 C y ¼ 7 D y ¼ 6

b aþ 12¼ 3

A a ¼ 4 B a ¼ 5 C a ¼ 6 D a ¼ 7

c x� 14þ 2 ¼ 10

A x ¼ 49 B x ¼ 37 C x ¼ 33 D x ¼ 3

Stage 5.2

Video tutorial

Equations withalgebraic fractions

MAT10NAVT10026

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a 3y5¼ 9 b 2a

9¼ 2 c mþ 5

2¼ 6 d k � 2

5¼ 11

e nþ 53¼ �10 f y� 1

4¼ �2 g xþ 1

4þ 2 ¼ 10 h y� 1

5� 6 ¼ 3

i mþ 25� 1 ¼ 3 j x� 6

5þ 7 ¼ 0 k 2ðxþ 1Þ

5¼ 10 l 3ðm� 2Þ

4¼ 6

m 8ðnþ 1Þ3

þ 2 ¼ 4 n 5ð1� nÞ2

� 1 ¼ 3 o 4ð1þ dÞ3

þ 1 ¼ 7 13


a 2k3¼ 5

4b 3w

10¼ 2

5c 5x

2¼ � 10

3

d x� 12¼ xþ 1

4e yþ 2

5¼ y� 1

2f aþ 5

3¼ a� 1

8

g pþ 25¼ p� 5

2h 2y� 1

5¼ yþ 1

4i 3yþ 2

3¼ 2yþ 1

4

j w5þ w

2¼ 7 k w

2� w

5¼ 15 l 2w

3� w

4¼ 4

m 3a2þ a

3¼ 1 n 2y

5� y

3¼ 4 o a

3þ 3a

4¼ 2

p 2m5� m

10¼ 1 q 4h

3þ h

5¼ 3 r 5y� 2

7¼ 3yþ 5

34 Solve each equation. Select the correct answer A, B, C or D.

a 4m5� m

3¼ 2

A m ¼ 10 B m ¼ 12 C m ¼ 307

D m ¼ 43

b mþ 12¼ 3þ 2m

5A m ¼ 1 B m ¼ 5 C m ¼ 5

3D m ¼ 2

3

6-03 Quadratic equations x2 þ bx þ c ¼ 0

An equation in which the highest power of the variable is 2 is called a quadratic equation; forexample, x2 ¼ 5, 3m2 þ 7 ¼ 10, d2 � d � 6 ¼ 0 and 4y2 � 3y ¼ 8.

Stage 5.2

See Example 4

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Solving ax2 ¼ c

Summary

The quadratic equation x2 ¼ c (where c is a positive number) has two solutions,

x ¼ �ffiffifficpðwhich means x ¼

ffiffifficp

and x ¼ �ffiffifficpÞ

Example 5

Solve each quadratic equation.

a m2 ¼ 16 b 3x2 ¼ 75 c 3m2 � 12 ¼ 0

Solutiona m2 ¼ 16

m ¼ �ffiffiffiffiffi16p

¼ �4

Finding the square root of both sides.

b 3x2 ¼ 75

x2 ¼ 753

x2 ¼ 25

x ¼ �ffiffiffiffiffi25p

¼ �5

Finding the square root of both sides.

c 3m2 � 12 ¼ 0

3m2 � 12þ 12 ¼ 0þ 12

3m2 ¼ 12

m2 ¼ 123

m2 ¼ 4

m ¼ �ffiffiffi4p

¼ �2

Stage 5.2

Worksheet

Equations review

MAT10NAWK10043

Video tutorial

Simple quadraticequations

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Example 6

Solve 7x2 � 88 ¼ 0, writing the solution correct to one decimal place.

Solution7x2 � 88 ¼ 0

7x2 ¼ 88

x2 ¼ 887

x ¼ �ffiffiffiffiffi887

r

x ¼ �3:54562 . . .

� �3:5

Solving x2 þ bx þ c ¼ 0 by factorisingTo solve quadratic equations of the form x2 þ bx þ c ¼ 0, we need to factorise the quadraticexpression on the LHS, which we learnt in Chapter 4, Algebra.

Example 7

Solve x2 þ 5x þ 6 ¼ 0.

SolutionTo factorise x2 þ 5x þ 6, find two numbers that have a sum of 5 and a product of 6.The correct numbers are 2 and 3.

x2 þ 5xþ 6 ¼ 0

xþ 2ð Þ xþ 3ð Þ ¼ 0

The LHS has been factorised into two factors, (x þ 2) and (x þ 3), whose product is 0.If two numbers have a product of 0, then one of the numbers must be 0.) xþ 2 ¼ 0

) x ¼ �2oror

xþ 3 ¼ 0

x ¼ �3[ The solution to x2 þ 5x þ 6 ¼ 0 is x ¼ �2 or x ¼ �3.Check:

When x ¼ �2,LHS ¼ (�2)2 þ 5 3 (�2) þ 6 ¼ 0RHS ¼ 0LHS ¼ RHSWhen x ¼ �3,LHS ¼ (�3)2 þ 5 3 (�3) þ 6 ¼ 0RHS ¼ 0LHS ¼ RHS.

Stage 5.2

Video tutorial

Quadratic equations byfactorising

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Summary

When solving quadratic equations by factorising, the following property is used.If pq ¼ 0, then p ¼ 0 or q ¼ 0.

Example 8

Solve each quadratic equation.

a x2 � x � 2 ¼ 0 b u2 þ 3u � 28 ¼ 0c a2 � 2a ¼ 0 d w2 � 10w þ 25 ¼ 0

Solutiona x2 � x � 2 ¼ 0

Find two numbers that have a sum of �1 and a product of �2.They are �2 and 1.(x � 2)(x þ 1) ¼ 0) x� 2 ¼ 0

) x ¼ 2oror

xþ 1 ¼ 0

x ¼ �1[ The solution to x2 � x � 2 ¼ 0 is x ¼ 2 or x ¼ �1.Check:

When x ¼ 2,LHS ¼ 22 � 2 � 2 ¼ 0RHS ¼ 0LHS ¼ RHS.When x ¼ �1,LHS ¼ (�1)2 � (�1) � 2 ¼ 0RHS ¼ 0LHS ¼ RHS.

b u2 þ 3u � 28 ¼ 0Find two numbers that have a sum of 3 and a product of �28.They are 7 and �4(u þ 7)(u � 4) ¼ 0) uþ 7 ¼ 0

) u ¼ �7oror

u� 4 ¼ 0

u ¼ 4[ The solution to u2 þ 3u � 28 ¼ 0 is u ¼ �7 or u ¼ 4.

c a2 � 2a ¼ 0This requires a simpler factorisation as there are only two terms, both involving a.a(a � 2) ¼ 0) a ¼ 0

) a ¼ 0oror

a� 2 ¼ 0

a ¼ 2[ The solution to a2 � 2a ¼ 0 is a ¼ 0 or a ¼ 2.

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d w2 � 10w þ 25 ¼ 0Find two numbers that have a sum of �10 and a product of 25.They are �5 and �5.(w � 5)(w � 5) ¼ 0) w� 5 ¼ 0

) w ¼ 5oror

w� 5 ¼ 0

w ¼ 5[ The equation w2 � 10w þ 25 ¼ 0 has only one solution, w ¼ 5.

Exercise 6-03 Quadratic equations x2 þ bx þ c ¼ 0

1 Solve each quadratic equation.

a m2 ¼ 144 b x2 ¼ 400 c y2 ¼ 225 d k2 � 169 ¼ 0e y2 � 1 ¼ 0 f w2 � 16 ¼ 0 g x2 þ 10 ¼ 14 h t2 � 9 ¼ 7

i a2

2¼ 8 j 5k2 ¼ 180 k 3w2 ¼ 300 l d2 þ 60 ¼ 204

m k2

2¼ 0:5 n w2

10¼ 2:5 o 4x2 ¼ 1 p m2

4¼ 9

q 5y2 ¼ 5 r 2p2 þ 3 ¼ 21 s 3k2

4þ 5 ¼ 8 t y2

5� 2 ¼ 18

2 Solve each equation (correct to one decimal place, where necessary).

a 5m2 � 20 ¼ 0 b 4a2

9¼ 36 c m2 ¼ 28 d 9m2 � 2 ¼ 32

e 9k2 þ 10 ¼ 13 f 2x2

5¼ 23 g k 2

16¼ 6 h 3k 2

10¼ 27

i 6y2 ¼ 0.726 j 3a2 þ 11 ¼ 267 k 2y2 � 14 ¼ 63 l 2w2

5� 1 ¼ 19

3 Solve each quadratic equation. Select the correct answer A, B, C or D.a x2 ¼ 121

A x ¼ 12, �12 B x ¼ 11, �11 C x ¼ 10, 11 D x ¼ 12, �11

b 9m2 � 1 ¼ 35

A m ¼ 3, �3 B m ¼ 2, �2 C m ¼ 8, �8 D m ¼ 9, �9

4 Solve the following.

a x2 þ 3x þ 2 ¼ 0 b y2 þ 5y þ 4 ¼ 0 c y2 þ 16y þ 48 ¼ 0d x2 þ x � 12 ¼ 0 e x2 þ 2x � 3 ¼ 0 f x2 þ 3x � 40 ¼ 0

5 Solve the following.

a x2 � x � 30 ¼ 0 b x2 � 8x þ 16 ¼ 0 c x2 � 5x � 66 ¼ 0d d2 � 2d ¼ 0 e x2 � 3x � 10 ¼ 0 f n2 þ 4n ¼ 0g k2 � 7k ¼ 0 h y2 ¼ 5y i v2 ¼ 12v

6 Explain why the quadratic equation x2 ¼ �25 has no solutions.

7 State which of these quadratic equations have no solutions. Give reasons.

a x2 ¼ �9 b 2k2 þ 5 ¼ 9 c 3m2 þ 8 ¼ 4

d 9w2

2� 1 ¼ 1 e 4þ d2

3¼ 8 f 5a2

2þ 3 ¼ 2

Stage 5.2

See Example 5

See Example 6

See Example 7

See Example 8

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8 Solve each quadratic equation. Select the correct answer A, B, C or D.a x2 þ 4x � 60 ¼ 0

A x ¼ �10, 6 B x ¼ 12, �5 C x ¼ 10, �6 D x ¼ �12, 5

b q2 þ 3q ¼ 0

A q ¼ 3, �3 B q ¼ 6, �3 C q ¼ 0, �3 D q ¼ 0, 3

6-04 Equation problems

Example 9

A rectangle is four times as long as it is wide. The perimeter of the rectangle is 180 mm.Find the dimensions of the rectangle.

SolutionLet the width of the rectangle be w mm. Then the length is 4w mm.

4w mm

w mm

) Perimeter: wþ 4wþ wþ 4w ¼ 180

10w ¼ 180

w ¼ 18010

¼ 18

[ The width of the rectangle is 18 mm and its length is 4 3 18 ¼ 72 mm.Check: Perimeter ¼ 18 þ 72 þ 18 þ 72 ¼ 180 mm.

Example 10

Jennifer is 7 years older than Amy. Ten years from now, the sum of their ages will be 43.How old are they now?

SolutionLet x ¼ Amy’s age now.Then Jennifer’s age now ¼ x þ 7.Break the information into ‘Now’ and ‘In 10 years time’

Now In 10 years timeAmy x x þ 10Jennifer x þ 7 x þ 7 þ 10 ¼ x þ 17

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In 10 year’s time:

Sum of ages: ðxþ 10Þ þ ðxþ 17Þ ¼ 43

2xþ 27 ¼ 43

2x ¼ 16

x ¼ 8

Amy is 8 years old now and Jennifer is 8 þ 7 ¼ 15 years old now.Check: In 10 years time, the sum of their ages will be 18 þ 25 ¼ 43.

Exercise 6-04 Equation problemsFor each question, write an equation and solve it to answer the problem.1 The longer sides of an isosceles triangle are twice as long as the shorter side. The perimeter of

the triangle is 90 mm. Find the lengths of the sides of the triangle.

2 The length of a rectangle is three times as long as its width. The perimeter of the rectangle is152 mm. Find its dimensions.

3 The length of a rectangle is three more than twice its width. Find the dimensions of therectangle if its perimeter is 84 cm.

4 The sum of three consecutive integers is 186. Find the integers.

5 A father is nine times the age of his son. In 5 years, he will be four times the age of his son.How old are they now?

6 When 15 is subtracted from three times a certain number, the answer is 63. What is thenumber?

7 The product of 2 and a number is the same as 12 minus the number. Find the number.

8 The sum of the present ages of Vatha and Chris is 36. In 4 years time, the sum of their ageswill equal twice Vatha’s present age. How old are they now?

9 Four consecutive integers have a sum of 858. Find the four integers.

10 Find the size of x in the diagram below.

(2x + 45)°

5(x – 12)°

See Example 9

See Example 10

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11 Manori’s bag contained 10-cent and 20-cent coins. She had 202 coins, with a total value of$31.90. How many 20-cent coins did Manori have?

12 If 17 more than a number is 5 more than three times the number, what is the number?

13 The sum of Scott’s age and his mother’s age is 45. In 5 years time, three times Scott’s age less 9will be the same as his mother’s age. Find the present ages of Scott and his mother.

14 One angle in a triangle is double the smallest angle, and the third angle in the triangle is 5more than four times the smallest angle. Find the size of each angle.

Mental skills 6 Maths without calculators

Multiplying and dividing by 5, 15, 25 and 50It is easier to multiply or divide a number by 10 than by 5. So whenever we multiply ordivide a number by 5, we can double the 5 (to make 10) and then adjust the first number.

1 Study each example.

a To multiply by 5, halve the number, then multiply by 10.

18 3 5 ¼ 18 312

3 10 ðor 9 3 2 3 10Þ

¼ 9 3 10

¼ 90

b To multiply by 50, halve the number, then multiply by 100.

26 3 50 ¼ 26 312

3 100 ðor 13 3 2 3 100Þ

¼ 13 3 100

¼ 1300

c To multiply by 25, quarter the number, then multiply by 100.

44 3 25 ¼ 44 314

3 100 ðor 11 3 4 3 25Þ

¼ 11 3 100

¼ 1100

d To multiply by 15, halve the number, then multiply by 30.

8 3 15 ¼ 8 312

3 30 ðor 4 3 2 3 15Þ

¼ 4 3 30

¼ 120

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e To divide by 5, divide by 10 and double the answer. We do this because there are two5s in every 10.

140 4 5 ¼ 140 4 10 3 2

¼ 14 3 2

¼ 28

f To divide by 50, divide by 100 and double the answer. This is because there are two50s in every 100.

400 4 50 ¼ 400 4 100 3 2

¼ 4 3 2

¼ 8

g To divide by 25, divide by 100 and multiply the answer by 4. This is because there arefour 25s in every 100.

600 4 25 ¼ 600 4 100 3 4

¼ 6 3 4

¼ 24

h To divide by 15, divide by 30 and double the answer. This is because there are two15s in every 30.

240 4 15 ¼ 240 4 30 3 2

¼ 8 3 2

¼ 16

2 Now evaluate each expression.

a 32 3 5 b 14 3 5 c 48 3 5 d 18 3 50e 52 3 50 f 36 3 25 g 28 3 5 h 12 3 25i 12 3 15 j 22 3 35 k 90 4 5 l 170 4 5m 230 4 5 n 1300 4 50 o 900 4 50 p 300 4 25q 1000 4 25 r 360 4 45 s 210 4 15 t 360 4 15

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Stage 5.2 6-05 Equations and formulasA formula is an equation that describes a relationship between variables. For example, the formulafor the perimeter of a rectangle is P ¼ 2(l þ w), where P is the perimeter, l is the rectangle’s lengthand w is the width. Because the formula is for the perimeter, P is called the subject of the formulaand it is the variable on its own on the left side of the ‘¼’ sign.

Example 11

The cost, C, in dollars, of hiring a taxi is C ¼ 5 þ 2.4d, where d is the distance travelled,in kilometres.

a Find the cost of a taxi trip if the distance travelled is 15 km.b Find the distance travelled by the taxi if the cost of the trip was $78.20.

Solutiona When d ¼ 15:

C ¼ 5þ 2:4 3 15

¼ 41The cost was $41.

b When C ¼ 78.20:78:20 ¼ 5þ 2:4d

73:20 ¼ 2:4d

d ¼ 73:202:4

¼ 30:5The distance travelled was 30.5 km.

Example 12

The surface area of a sphere is given by the formula A ¼ 4pr2, where r is the radius.Find (correct to one decimal place):a the surface area of a sphere with radius 2.8 cmb the radius of a sphere with surface area 40 m2.

Puzzle sheet

Getting the rightformula

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Solutiona When r ¼ 2.8:

A ¼ 4 3 p 3 2:82

¼ 98:520 . . .

� 98:5 cm2

b When r ¼ 40:40 ¼ 4pr2

4pr2 ¼ 40

r2 ¼ 404p

¼ 3:183 . . .

r ¼ffiffiffiffiffiffiffiffiffiffiffi3:183p

¼ 1:784 . . .

� 1:8 m

Since r > 0, the radius is positive.

Exercise 6-05 Equations and formulas1 The formula for the perimeter of a rectangle is P ¼ 2(l þ w).

a Find the perimeter of a rectangle with length 10 cm and width 16 cm.

b Find the width of a rectangle whose perimeter is 58 m and length is 12 m.

2 A formula for converting speed expressed in m/s (metres/second) to a speed expressed inkm/h is k ¼ 3.6M, where M is the speed in m/s. Convert each speed to km/h.

a 10 m/s b 24 m/s c 50 m/s

3 A car is travelling at a speed of 110 km/h on a freeway. Use the formula from question 2 tocalculate how fast this is in m/s.

4 The average of two numbers, m and n, is A ¼ mþ n2

. If two numbers have an average of 28

and one of the numbers is 13, use the formula to find the other number.

5 The formula for converting temperature recorded in �F to temperature in �C is

C ¼ 59ðF � 32Þ. Express each temperature in �C, correct to the nearest degree.

a 80�F b 32�F c 212�F d 102�F

6 Pythagoras’ theorem for a right-angled triangle with sides a, b and c (the hypotenuse) isc2 ¼ a2 þ b2. Find, correct to one decimal place where necessary:

a c, if a ¼ 5 and b ¼ 10 b a, if c ¼ 41 and b ¼ 40 c b, if c ¼ 20 and a ¼ 10

7 The formula for the circumference of a circle is C ¼ 2pr, where r is the radius. Find, correct toone decimal place:a the circumference of a circle with radius 2.4 m

b the radius of a circle whose circumference is 200 cm.

Stage 5.2

See Example 11

See Example 12

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8 The body mass index (BMI) of an adult is B ¼ Mh2, where M is the mass in kilograms and h is

the height in metres. Find, correct to one decimal place:a the BMI of Dean who is 1.85 m tall and has a mass of 72 kg

b the mass of a person with a BMI of 24, who is 2.1 m tall.

9 The volume of a sphere is V ¼ 43

pr3, where r is the radius. Find, correct to one decimal place:a the volume of a sphere with radius 3.2 cm

b the radius of a sphere with a volume of 500 m3.

10 The average speed in km/h of a car is given by the formula S ¼ DT

, where D is the distancecovered in kilometres and T is the time taken in hours. Find, correct to the nearest whole number:a the average speed of a car that takes 4.5 hours to travel a distance of 420 km

b the distance travelled, if a car maintains a speed of 87.2 km/h for 5 hours

c the time taken, if a distance of 650 km is covered at a speed of 91 km/h.

11 The cost, C, (in dollars) of hiring a car is C ¼ 45 þ 0.15d, where d is the number of kilometrestravelled. Calculate:a the cost of hiring a car to travel 350 km

b the distance travelled, if the cost is $138.

12 The surface area of a closed cylinder is given by the formula SA ¼ 2pr2 þ 2prh. Calculate,correct to one decimal place:a the surface area of a cylinder with radius 2.1 m and height 3.5 m

b the height of a cylinder with surface area 1255.38 cm2 and radius 9 cm.

6-06 Graphing inequalities on a number lineAn inequality looks like an equation except that the equals sign (¼) is replaced by an inequalitysymbol >, �, < or �.2x � 7 ¼ 15 is an equation. There is only one value of x that makes it true.2x � 7 � 15 is an inequality. There is a range of values of x that make it true.

Summary

> means ‘is greater than’ � means ‘is greater than or equal to’< means ‘is less than’ � means ‘is less than or equal to’

The inequality x � 3 is read ‘x is greater than or equal to 3’. It includes 3 and all the numbersabove 3, such as 3.01, 4, 10, 20 000, etc.The inequality x > 3 is read ‘x is greater than 3’ and means all the numbers above 3, but not 3.

Inequality In words Meaningx > 3 x is greater than 3 Values above 3x < 3 x is less than 3 Values below 3x � 3 x is greater than or equal to 3 Values above and including 3x � 3 x is less than or equal to 3 Values below and including 3

For convenience, we can represent all the values in an inequality using a number line.

Stage 5.2

Worksheet

Graphing inequalities

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Stage 5.2Example 13

Graph each inequality on a number line.

a x � 1 b x < 5 c x > �3

Solutiona x � 1 means that x can be any number greater than 1 or equal to 1.

–3 –2 –1 0 1 2 3 4 5 6x

b x < 5 means that x can be any number less than 5, but not including 5.

–3 –2 –1 0 1 2 3 4 5 6x

c x > �3 means that x can be any number greater than �3, but not including �3.

–3 –2 –1 0 1 2 3 4 5 6x

Exercise 6-06 Graphing inequalities on a number line1 Graph each inequality on a separate number line.

a x � 2 b x < �3 c x � 1 d x > 7e x � 4 f x > 0 g x � �2 h x < 10

2 Write the inequality illustrated by each number line.

a–2 0 2 4 6

x b0 1 2 3 4 5 6

x

c–8–10 –6 –4 –2 0

x d–8–10 –6 –4 –2 0 2

x

3 Which inequality is graphed below? Select the correct answer A, B, C or D.

–7 –6 –5 –4 –3 –2 –1 0 1 2x

A x > �2.5 B x < �2.5 C x < �3.5 D x > �3.5

4 Write the inequality represented on each number line.

a–3 –2 –1 0 1 2 3 x

b–1 0 1 2 3 4 5 x

c0 2 4 6 8 10 12 x

d–3 –2 –1 0 1 2 3 x

e–9 –6 –3 0 3 6 9 x

f–3 –2 –1 0 1 2 3 x

g–10 –8 –6 –4 –2 0 2 x

h–5 0 5 10 15 20 25 x

i–1 0 1 2 3 4 5 x

The filled circle at 1 means weinclude 1.

The open circle on 5 meansthat 5 is not included

See Example 13

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Investigation: The language of inequalities

Work in pairs to complete this activity.Use inequality symbols to write each statement algebraically.a The minimum height (H) for rides at an amusement park is 1.3 m.b The speed limit in a school zone is 40 km/h.c To be eligible to vote, you must be at least 18 years old (A ¼ age).d The overseas tour is only for people whose age (A) is from 18 to 35.e The cost (A) of a tennis racquet will be at least $95 but no more than $360.f A new flute (F) costs at least $475.g The price of units (U) in a new block start at $240 000.

Stage 5.2

Investigation: Solving inequalities

We have solved equations by doing the same thing to both sides (keeping the equation‘balanced’). Will this method work with inequalities, such as x þ 4 > 10 or 6x < 13?1 Start with an inequality that is true, such as 7 > 4.2 Add 5 (or any number you choose) to both sides of the inequality; for example 7 > 4

becomes 12 > 9. Is the new inequality true or false?3 Subtract 9 (or any number you choose) from each side of the original inequality; for

example 7 > 4 becomes �2 > �5. Is the new inequality true or false?4 Multiply both sides of the original inequality by 4 (or any positive number you choose);

for example 7 > 4 becomes 28 > 16. Is the new inequality true or false?5 Divide both sides of the original inequality by 2 (or any positive number you choose);

for example 7 > 4 becomes 3 12 > 2. Is the new inequality true or false?

6 Multiply both sides of the original inequality by �3 (or any negative number you choose);for example 7 > 4 becomes �21 > �12. Is the new inequality true or false?

7 Divide both sides of the original inequality by �4 (or any negative number you choose),for example 7 > 4 becomes �1 3

4 > �1. Is the new inequality true or false?8 Which of the six operations used in questions 2 to 7 can be used on inequalities to give a

true result?9 Which of the six operations used in questions 2 to 7 cannot be used with inequalities

because they give a false result?10 Copy and complete the following inequality statements.

a 6 < 86 3 3 < 8 3 ___ (multiplying both sides by 3)[ 18 __ 24

b 10 > �410 4 2 __ �4 4 __ (dividing both sides by 2)[ __________

Does the inequality sign (< or >) stay the same when multiplying or dividing by a positivenumber?

11 a Is it true that 5 < 8?b Multiply both sides by �2. Is it true that �10 < �16?c What needs to be reversed to change �10 < �16 into a true inequality statement?d Copy and complete the following to make a true inequality statement: �10 ______ �16.

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6-07 Solving inequalities

Example 14

Solve each inequality and graph its solution on a number line.

a 2x � 10 � 16 b 2(y � 1) � 12 c wþ 32

> �1

Solutiona 2x� 10 � 16

2x� 10þ 10 � 16

2x � 262x

2� 26

2x � 13

10 11 12 13 14 15x

b 2ðy� 1Þ � 12

2y� 2 � 12

2y� 2þ 2 � 12þ 2

2y � 142y

2� 14

2y � 7

0 1 2 3 4 5 6 7 8 10 119y

c wþ 32

> �1

wþ 32

3 2 > �1 3 2

wþ 3 > �2

wþ 3� 3 > �2� 3

w > �5

–6 –5 –4 –3 –2 –1 0 1w

Stage 5.212 a Is it true that 18 > �6?b Divide both sides by �3. Is it true that �6 > 2?c What needs to be reversed to change �6 > 2 into a true inequality statement?d Copy and complete the following to make a true inequality statement: �6 ____ 2.

13 Copy and complete:When multiplying or d__________ both sides of an inequality by a n__________ number,the inequality sign must be r__________.

Worksheet

Inequalities review

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Summary

Inequalities can be solved algebraically in the same way as equations, by using inverseoperations. However, when multiplying or dividing both sides of an inequality by a negativenumber, you must reverse the inequality sign.

Example 15

Solve each inequality.

a 1 � 2x � �11 b 4 � r < 7 c aþ 5�3

> 4

Solutiona 1� 2x � �11

1� 2x� 1 � �11� 1

�2x � �12�2x

�2� �12�2

x � 6

Dividing both sides by a negative numberreverses the inequality sign.

b 4� r < 7

4� r � 4 < 7� 4

�r < 3�r

�1>

3�1

r > �3

Dividing both sides by a negative numberreverses the inequality sign.

c aþ 5�3

> 4

aþ 5�3

3 �3ð Þ < 4 3 �3ð Þ

aþ 5 < �12

aþ 5� 5 < �12� 5

a < �17

Multiplying both sides by a negativenumber reverses the inequality sign.

Exercise 6-07 Solving inequalities1 Solve each inequality and graph its solution on a number line.

a x � 1 > 6 b 3y � 12 c m þ 4 � 2d x

5� �20 e 12x < 60 f 5y > �20

g 4a � 2 h 3w � �30 i 8a þ 5 � 45j 3a þ 1 � 10 k 6a þ 4 � �2 l 3w � 3 < �12m 5a þ 3 � �27 n 5y þ 1 � 16 o 4a þ 5 < 15

Stage 5.2

See Example 14

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2 What is the solution to 3a � 3 > �18? Select the correct answer A, B, C or D.

A a > �5 B a > �7 C a > 5 D a > 9

3 Solve each inequality.

a 3(x þ 2) � 9 b 5(m � 4) � 10 c 2(y þ 5) � �6d 3(w � 2) > �6 e 5(2w þ 3) � 15 f 4(2m � 5) � 8

g mþ 53� 1 h x� 1

2� 2 i w� 2

5> �1

j 2aþ 13

< 3 k 5aþ 24� 8 l 2ðmþ 1Þ

3� 3

m 5ðm� 1Þ4

> 3 n 4ðm� 2Þ3

� �6 o 3þ x5< 10

4 What is the solution to x� 25� �1? Select A, B, C or D.

A x � �7 B x � �3 C x � 10 D x � �3

5 Solve each inequality and graph its solution on a number line.

a 5 � x � 2 b 15 > 7 � y c 1 � k < 12d 7 � m � 7 e 2 � p > 8 f �t þ 6 � 10


a �2x < 6 b k�3� 4 c �5t > 12

d �x3� �4 e 4 � 3w > 7 f �4y þ 3 � 11

g 3 � 2x � �5 h 8 � 5a < 3 i �2d � 3 > 8

j 5þ w�3

> 2 k x� 4�4� 3 l �pþ 2

�3< �2

Just for the record Film and game classificationIn Australia, films, publications and computer games are rated by the Classification Board. Filmsand videos are rated G, PG, M, MA15þ or R18þ, with each category containing a list ofguidelines related to the film’s use of violence, coarse language, adult themes, sex and nudity.General (G) means suitable for all ages. Children can watch films classified G without adultsupervision.

Parental guidance (PG) means that parental guidance is recommended for persons under 15years of age. These films contain material that may be confusing or upsetting to children, butnot harmful or disturbing. Parents should watch the film with their children or preview it tocheck elements such as language used or inappropriate themes.

Stage 5.2

See Example 15

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Mature (M) means recommended for mature audiences, 15 years and over. The film orcomputer game may contain material that is harmful or disturbing to children, but the impactis not so strong as to require restriction.

Mature accompanied (MA15þ) means legal restrictions apply to persons under the age of 15.Children are not allowed to see MA films unless accompanied by a parent or guardian,because they contain material that is likely to be harmful or disturbing to them.

Restricted (R18þ) means legally restricted to adults, 18 years and over. It applies to films thatdeal with issues and scenes that require an adult perspective, and so are unsuitable for personsunder 18 years of age. A person will be asked for proof of age before buying, hiring orviewing films or computer games in this category.

1 Each of the classifications is represented by a logo (as shown) with the letter inside aparticular shape. What shape is each logo?2 Write each classification category as an inequality.

Power plus


a 3ð1� yÞ5

¼ 4� 2yb 50

2y¼ 10 c 2mþ 5

3� 1 ¼ mþ 4

2 If y ¼ abþ cde

, find d if y ¼ �12, a ¼ �1, b ¼ �8, c ¼ 7 and e ¼ 4.

3 James is ten years older than Brett. In three years time, James will be twice as old as Brett.How old are James and Brett now?

4 One third of a number added to one-sixth of a number is 18. What is the number?

5 Graph each inequality on a number line.

a 1 � x � 4 b �2 � x � 3 c �12 < 4x � 4

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Chapter 6 review

n Language of maths

brackets check equation expand

factorise formula fraction greater than

inequality inverse operation LHS less than

lowest common multiple (LCM) number line quadratic equation RHS

solution solve square root subject

substitute variable

1 What type of equation has 2 as the highest power of x? Write an example of this type ofequation.

2 True or false? 10 � 10.

3 What is the difference between an equation and an inequality?

4 Why is it possible for a quadratic equation to have more than one solution?

5 When checking the solution to an equation, you need to show that ‘LHS ¼ RHS’. What doesthat mean?

6 What does the inequality symbol ‘�’ mean?

n Topic overview

Copy and complete the table below.

The best part of thischapter was …

The worst part was …

New work consisted of …

?

I need help with …

Puzzle sheet

Equations andinequalities crossword

MAT10NAPS10047

Quiz

Equations

MAT10NAQZ00011

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Copy and complete this mind map of the topic, adding detail to its branches and usingpictures, symbols and colour where needed. Ask your teacher to check your work.

Quadratic equations

Graphinginequalities on

a number of lineEquations

and formulas

Equation problems

Equation withalgebraic fractions

Equations

Solvinginequalities

Equations andinequalities

9780170194655212

Chapter 6 review


a 3a þ 10 ¼ 43 b 8y þ 5 ¼ 2y þ 21 c 2a � 12 ¼ 6a

d 9 � 2y ¼ 5 þ 2y e 3(m � 2) ¼ 27 f 2(2a þ 1) ¼ 3(a þ 10)g 5(h þ 1) þ 3(h � 2) ¼ 12 h 4(2y þ 1) þ 3(1 þ 4y) ¼ 20


a 3wþ 25¼ 4 b y

5¼ 7

4c 2aþ 1

2¼ 3a� 1

4

d 3mþ 56¼ 10� m

3e 2s

3� s

6¼ 2 f x

10þ x

2¼ 1

3 What is the solution to2ðp� 1Þ

3¼ 4? Select the correct answer A, B, C or D.

A p ¼ 7 B p ¼ 11 C p ¼ 5 D p ¼ 4

4 Solve each quadratic equation.

a y2 ¼ 4 b p2 � 100 ¼ 0 c 4x2 ¼ 36

d 3m2 � 3 ¼ 0 e 2w2

5¼ 10 f x2 þ 8x þ 7 ¼ 0

g h2 � 8h � 9 ¼ 0 h u2 þ 4u � 77 ¼ 0 i k2 þ 5k ¼ 0j m2 � 2m ¼ 0 k b2 þ 20b þ 100 ¼ 0 l w2 ¼ 9w

5 The sum of four consecutive numbers is 374. Find the four numbers.

6 Find the value of x.

(3x – 20)°

(x + 18)°

7 The braking distance (in metres) of a bicycle travelling at a speed of v metres/second is

d ¼ vðvþ 1Þ2

. Calculate the braking distance when the speed of the bicycle is 15 m/s.

8 The volume of a pyramid is given by the formula V ¼ 13

Ah, where A is the area of the baseand h is the perpendicular height of the pyramid. Find:a the volume of a pyramid with a base area of 48 mm2 and a perpendicular height of 10 mmb the base area of a pyramid with a volume of 500 m3 and a perpendicular height of 5 m

9 Graph each inequality on a number line.

a x � 0 b x < 3 c x � �2 d x > �5


a y� 6 � 10 b 2y � �15 c 3a þ 10 > �5

d 10 � 6x < 28 e aþ 2�4

>72

f 3� 5x2� 9

See Exercise 6-01

Stage 5.2

See Exercise 6-02

See Exercise 6-02

See Exercise 6-03

See Exercise 6-04

Stage 5.2

See Exercise 6-05

See Exercise 6-06

See Exercise 6-07

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Chapter 6 revision

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