7 stretch lesson: equations and inequalities images... · equations and inequalities 7.1 linear...
TRANSCRIPT
Stretch objectives
Before you start this chapter, mark how confi dent you feel about each of the statements below:
I can solve linear equations involving fractions.
I can solve quadratic equations by factorising.
I can solve two inequalities and compare them to fi nd values that satisfy both inequalities.
Check-in questions
• Complete these questions to assess how much you remember about each topic. Then mark your work using the answers at the end of the lesson.
• If you score well on all sections, you can go straight to the Revision Checklist and Exam-style Questions at the end of the lesson. If you don’t score well, go to the lesson section indicated and work through the examples and practice questions there.
1 Solve the equation 3 13
x − = 4 + 2x Go to 7.1
2 Solve these quadratic equations. Go to 7.2
a x2 - 7x = 0 b x2 + 8x + 15 = 0 c x2 - 5x + 6 = 0
3 a Solve the inequality 4 + x > 7x - 8
b Solve the inequality 3 54x + 5. Represent the solutions
on a copy of the number line. Go to 7.2
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
7 Stretch lesson: Equations and inequalities
7.1 Linear equations with fractionsWhen equations involve fractions, multiply both sides by the denominator to eliminate the fraction part of the equation.
Solve: x + 43
= 10
x + 4 = 30
x = 26Multiply both sides by 3.
Subtract 4 from both sides.
Example
1Q
A
AQA GCSE (9-1) Maths for Post-16 © HarperCollinsPublishers 2017
Solve: x + 23
+ x − 12
= 156
2(x + 2) + 3(x - 1) = 15
2x + 4 + 3x - 3 = 15
5x + 1 = 15
5x = 15 - 1 x = 14
5 or 2 45
Example
3Q
A 6 is the lowest common multiple of 2, 3 and 6, so multiply both sides of the equation by 6.
Expand the brackets.
Solve.
You can write your answer as an improper fraction, a mixed number or an exact decimal.
Exam tips Make sure that you write down each step in the solution.
Practice questions 1 Solve these equations.
a x + =65 2 b x − =3
2 5 c x + =164 6
2 Solve these equations.
a 92 3− =x b 15 2
3 3− =x c 29 35 7− =x
3 Solve these equations.
a x x+x x+x x+x x+x x − =3x x3x x2
14 5 b x x+x x+x x+x x+x x − =6x x6x x
52
10115
c 2 12
34
154
x x2 1x x2 12 1+2 12 1x x2 1+2 1x x2 1 +x x+x x + =
4 Tzun is asked to solve 2 68 162 6x2 62 6+2 6 = .
This is his working: 28 = 16 – 6x
28 = 10x
2x = 10 − 8
2x = 2
x = 1
Identify where Tzun went wrong and work out the correct value for x.
Solve: 3 2
5( )3 2( )3 2 1( )1x( )x −( )−
= 6
3(2x - 1) = 6 × 5
6x - 3 = 30
6x = 33
x = 336
x = 5.5
First, multiply both sides by 5.
Example
2Q
A
AQA GCSE (9-1) Maths for Post-16 © HarperCollinsPublishers 2017
7.2 Quadratic equationsA quadratic equation written in the form ax2 + bx + c = 0 can be solved by factorising into two brackets (x ± ?)(x ± ?) = 0. (See Chapter 6 for more on factorisation.)
Since the equation equals zero, at least one of the brackets must equal zero.
To solve the equation x2 - x - 6 = 0:
• Factorise into two brackets. (x + 2)(x - 3) = 0
• Either (x + 2) = 0 or (x - 3) = 0
So x = -2 or x = 3
Solve: x2 - 7x + 10 = 0
(x - 2)(x - 5) = 0
Either (x - 2) = 0 or (x - 5) = 0
So x = 2 or x = 5
Example
4Q
A
Solve: x2 - 6x - 16 = 0
(x − 8)(x + 2) = 0
Either (x − 8) = 0 or (x + 2) = 0
So x = 8 or x = −2
Example
5Q
A
Exam tips Check that the equation is written in the form ax2 + bx + c = 0 before you factorise.
Practice questions 1 Factorise these quadratic expressions.
a x2 + 6x + 8 b x2 + 12x + 20 c x2 + 7x + 12 d x2 + 12x + 36
2 Use factorisation to solve these quadratic equations.
a x2 + 7x + 10 = 0 b x2 + 13x + 36 = 0
c x2 + 13x + 30 = 0 d x2 + 12x + 35 = 0
3 Solve these quadratic equations.
a x2 - x - 2 = 0 b x2 - 5x + 6 = 0
c x2 - 2x - 8 = 0 d x2 - 8x + 16 = 0
4 Solve these.
a x2 + 4x = −3 b x2 - x - 3 = 3 c x2 + 8x + 3 = −9
5 The area of the square is 64 cm². Find the value of x.
(x + 3) cm
AQA GCSE (9-1) Maths for Post-16 © HarperCollinsPublishers 2017
Solve: 3 24
x − < 4
3x - 2 < 16
3x < 16 + 2
3x < 18
x < 183
x < 6
Multiply both sides by 4.
Add 2 to both sides.
Divide both sides by 3.
Example
6Q
A
Solve: -7 < 3x - 1 11
-6 < 3x 12
-2 < x 4
The integer values that satisfy this inequality are -1, 0, 1, 2, 3 and 4.
Add 1 to each part of the inequality.
Divide each part of the inequality by 3.
Example
7Q
A
Solve: 2 < 2 53
x − < 5
6 < 2x - 5 < 15
11 < 2x < 20
5.5 < x < 10
The integer values that satisfy this inequality are 6, 7, 8 and 9.
Multiply each part of the inequality by 3.
Add 5 to each part of the inequality.
Divide each part of the inequality by 2.
Example
8Q
A
7.3 Further inequalitiesInequalities involving fractionsFollow the same process for dealing with inequalities involving fractions as you did with equations - multiply through to remove the denominator.
Two inequalitiesWhen there are two inequalities, make sure that you do the same thing to all parts of the inequality.
Practice questions 1 Solve these inequalities.
a 2 15 3x + > b x − <7
4 2 5. c 5 3
9x –
3 d 8 6
10x –
0.3
2 Solve these inequalities.
a 5 2x + 1 < 11 b −8 3x + 1 < 13 c 4 4x < 10 d −10 4x + 2 < 2
AQA GCSE (9-1) Maths for Post-16 © HarperCollinsPublishers 2017
Exam-style questions 1 Which integers satisfy 2 < 2x + 5 15?
2 Write down the largest integer which satisfies 2 5
4 2x– –< .
3 Solve: 5 2 40 53x x+( ) = −
4 Write an inequality for the integers that satisfy both of these inequalities.
−5 x 3 −2 < 2x + 2 8
5 Solve: x2 - 8x + 15 = 0
6 This rectangle has area 44 cm2. Find the length of the longest side.
(x – 4) cm
(x + 3) cm
7 Solve: 2x2 + 8x + 6 = 0
8 Solve: x2 - 7x + 6 = −6
9 The area x of a field is given as x2 + x - 12 = 0. Solve to find the value of x.
REVISION CHECKLIST ● Some quadratic equations can be solved by factorisation.
AQA GCSE (9-1) Maths for Post-16 © HarperCollinsPublishers 2017
Chapter 7 Stretch lesson: AnswersCheck-in questions
1 x = −4 132 a x = 0 or x = 7
b x = −5 or x = −3
c x = 2 or x = 3
3 a x < 2
b –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10
7.1 Linear equations with fractions
1 a x = 4 b x = 13 c x = 8
2 a x = 3 b x = 3 c x = −2
3 a x = 5 b x = 4 c x = 2
4 Tzun doesn’t eliminate the denominator first. He also subtracts 8 instead of multiplying by 8 in the third line.
The correct working is: 2x + 6 = 16 × 8
2x + 6 = 128
2x = 128 – 6
2x = 122
x = 122 ÷ 2
x = 61
7.2 Quadratic equations
1 a (x + 4)(x + 2) b (x + 10)(x + 2) c (x + 3)(x + 4) d (x + 6)(x + 6)
2 a x = –2 or x = –5 b x = –9 or x = –4 c x = –10 or x= –3 d x = –7 or x = –5
3 a x = 2 or x = –1 b x = 2 or x = 3 c x = 4 or x = –2 d x = 4
4 a x = –3 or x = –1 b x = 3 or x = –2 c x = –2 or x = –6
5 x = 5
7.3 Further inequalities
1 a x > 7 b x < 17 c x 6 d x 98
2 a 2 x < 5 b −3 x < 4 c 1 x < 2.5 d −3 x < 0
Exam-style questions
1 –1, 0, 1, 2, 3, 4 and 5
2 x = –2
3 x = 12
4 −2 < x 3
5 x = 3 or x = 5
6 11 cm
7 x = –1 or x = –3
8 x = 3 or x = 4
9 x = 3
AQA GCSE (9-1) Maths for Post-16 © HarperCollinsPublishers 2017