Equations and inequalities

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$each of the statements below: I can solve linear equations involving fractions. I can solve quadratic equations by factorising.$ I can solve two inequalities and compare them to find values that satisfy both inequalities.

I can solve two inequalities and compare them to find values that satisfy both inequalities.

Then mark your work using the answers 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 2 Solve these quadratic equations. Go to 7.

1 7 Stretch lesson: Equations and inequalities Stretch objectives Before you start this chapter, mark how confident 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 find 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 uestions 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 x 1 = + 2x Go to Solve these quadratic equations. Go to 7.2 a x 2-7x = 0 b x 2 + 8x + 1 = 0 c x 2 - x + 6 = 0 a Solve the inequality + x > 7x - 8 b Solve the inequality x +. Represent the solutions on a copy of the number line. Go to Linear equations with fractions When equations involve fractions, multiply both sides by the denominator to eliminate the fraction part of the equation. 1 Solve: x + x + = 0 x = 26 = 10 Multiply both sides by. Subtract from both sides.

2 2 ( ) Solve: (22x 1 (2x - 1) = 6 6x - = 0 = 6 First, multiply both sides by. 6x = x = 6 x =. Solve: x x 1 = (x + 2) + (x - 1) = 1 2x + + x - = 1 x + 1 = 1 x = 1-1 x = 1 or 2 6 is the lowest common multiple of 2, 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. x + 6 a = 2 b x = 2 c x + 16 = 6 2 Solve these equations. 9 x a = 2 b 1 2x = c 29 x = 7 Solve these equations. x+ x a 1 + = b 2 x+ 6 x 2 + = c 2 x+ 1 x + + = 2 1 Tzun is asked to solve 2 x + 6 = x This is his working: 8 =16 6 2x 8 =10 2x = x = 2 x = 1 Identify where Tzun went wrong and work out the correct value for x.

3 7.2 uadratic equations quadratic equation written in the form ax 2 + 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 x 2 - x - 6 = 0: Factorise into two brackets. (x + 2)(x - ) = 0 Either (x + 2) = 0 or (x - ) = 0 So x = -2 or x = Solve: x 2-7x + 10 = 0 (x - 2)(x - ) = 0 Either (x - 2) = 0 or (x - ) = 0 So x = 2 or x = Solve: x 2-6x - 16 = 0 (x 8)(x + 2) = 0 Either (x 8) = 0 or (x + 2) = 0 So x = 8 or x = 2 Exam tips Check that the equation is written in the form ax 2 + bx + c = 0 before you factorise. Practice questions 1 Factorise these quadratic expressions. a x 2 + 6x + 8 b x x + 20 c x 2 + 7x + 12 d x x Use factorisation to solve these quadratic equations. a x 2 + 7x + 10 = 0 b x 2 + 1x + 6 = 0 c x 2 + 1x + 0 = 0 d x x + = 0 Solve these quadratic equations. a x 2 - x - 2 = 0 b x 2 - x + 6 = 0 c x 2-2x - 8 = 0 d x 2-8x + 16 = 0 Solve these. a x 2 + x = b x 2 - x - = c x 2 + 8x + = 9 The area of the square is 6 cm². Find the value of x. (x + ) cm

4 7. Further inequalities Inequalities involving fractions Follow the same process for dealing with inequalities involving fractions as you did with equations - multiply through to remove the denominator. 6 Solve: x 2 < x - 2 < 16 x < x < 18 x < 18 x < 6 Multiply both sides by. dd 2 to both sides. Divide both sides by. Two inequalities When there are two inequalities, make sure that you do the same thing to all parts of the inequality. 7 Solve: -7 < x < x 12 dd 1 to each part of the inequality. -2 < x Divide each part of the inequality by. The integer values that satisfy this inequality are -1, 0, 1, 2, and. 8 Solve: 2 < 2 x < 6 < 2x - < 1 Multiply each part of the inequality by. 11 < 2x < 20 dd to each part of the inequality.. < x < 10 Divide each part of the inequality by 2. The integer values that satisfy this inequality are 6, 7, 8 and 9. Practice questions 1 Solve these inequalities. a 2x + 1 > b x 7 < 2. c x 9 d 8 x Solve these inequalities. a 2x + 1 < 11 b 8 x + 1 < 1 c x < 10 d 10 x + 2 < 2

5 REVISION CHECKLIST Some quadratic equations can be solved by factorisation. Exam-style questions 1 Which integers satisfy 2 < 2x + 1? 2 Write down the largest integer which satisfies 2 x < 2. Solve: 0 x ( x + 2)= Write an inequality for the integers that satisfy both of these inequalities. x 2 < 2x Solve: x 2-8x + 1 = 0 6 This rectangle has area cm 2. Find the length of the longest side. (x + ) cm (x ) cm 7 Solve: 2x 2 + 8x + 6 = 0 8 Solve: x 2-7x + 6 = 6 9 The area x of a field is given as x 2 + x - 12 = 0. Solve to find the value of x.

6 Chapter 7 Stretch lesson: nswers Check-in questions 1 x = 1 2 a x = 0 or x = 7 b x = or x = c x = 2 or x = a x < 2 b Linear equations with fractions 1 a x = b x = 1 c x = 8 2 a x = b x = c x = 2 a x = b x = c x = 2 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 = x + 6 = 128 2x = x = 122 x = x = uadratic equations 1 a (x + )(x + 2) b (x + 10)(x + 2) c (x + )(x + ) d (x + 6)(x + 6) 2 a x = 2 or x = b x = 9 or x = c x = 10 or x= d x = 7 or x = a x = 2 or x = 1 b x = 2 or x = c x = or x = 2 d x = a x = or x = 1 b x = or x = 2 c x = 2 or x = 6 x = 7. Further inequalities 1 a x > 7 b x < 17 c x 6 d x a 2 x < b x < c 1 x < 2. d x < 0 Exam-style questions 1 1, 0, 1, 2,, and 2 x = 2 x = < x x = or x = 6 11 cm 7 x = 1 or x = 8 x = or x = 9 x =

Edexcel GCSE (9-1) Maths for post-16

Edexcel GCSE (9-1) Maths for post-16 Fiona Mapp with Su Nicholson 227227_Front_EDEXCEL.indd 1 02/06/2017 11:34 Contents Introduction 5 How to use this book 13 1 Numbers 14 1.1 Positive and negative numbers