Inequalities — Paper 2 · IGCSE Mathematics 0580

4 Solve the inequality 5 −x3 < 17 . Answer [2]

2 marks

Mark scheme: 4 x > -4 or -4 < x 2* M1 -4 seen on answer line or M1 correct movement of 2 terms

This question in 0580/21 Oct/Nov 2004

20 For Examiner's y Use 6 5 4 3 2 1 0 x 1 2 3 4 5 6 –1 –2 –3 –4 –5 (a) Draw the three lines y = 4, 2x – y = 4 and x + y = 6 on the grid above. [4] (b) Write the letter R in the region defined by the three inequalities below. y Y 4 2x – y [ 4 x + y [ 6 [1]

5 marks

Mark scheme: 20 draw 2x – y = 4 2 W1 Line through (2,0) or (0,–4) draw x + y = 6 1 draw y = 4 1 R correct region identified by R 1 0 6

This question in 0580/21 May/June 2009

13 Solve the inequality 6(2 − 3x) − 4(1 − 2x)=Y= 0. For Examiner's Use Answer [3]

3 marks

Mark scheme: 13 x [ 0.8 or x [ 4 cao 3 B1 12 – 18x B1 –4 + 8x these terms may be 5 reversed if moved to the other side of the inequality allow >=

This question in 0580/21 Oct/Nov 2009

14 y 10 8 6 4 2 x –4 –2 0 2 4 6 8 10 12 14 16 18 By shading the unwanted regions of the grid above, find and label the region R which satisfies the following four inequalities. y [ 2 x + y [ 6 y Y x + 4 x + 2y Y 18 [4]

4 marks

Mark scheme: 14 4 Mark the position of the letter R (or the worst unshaded region if R is missing) as follows IGCSE – May/June 2010 0580 22

This question in 0580/22 May/June 2010

x − 1 For 13 Solve the inequality. 2x + 5 I Examiner's 4 Use Answer [3]

3 marks

Mark scheme: 13 x < –3 3 M1 correct move M1 correct move M1 correct move

This question in 0580/21 Oct/Nov 2010

20 For y Examiner's Use 8 7 6 5 4 3 2 1 x 0 1 2 3 4 5 6 7 8 (a) Draw the lines y = 2, x + y = 6 and y = 2x on the grid above. [4] (b) Label the region R which satisfies the three inequalities x + y [ 6, y [ 2 and y Y 2x. [1]

5 marks

Mark scheme: 20 (a) 4 B1 y = 2 single line thro B1 (6, 0) and B1 (0,6) R B1 y = 2x (b) 1 Correct R cao IGCSE – October/November 2010 0580 22

This question in 0580/22 Oct/Nov 2010

22 For y Examiner's Use 9 8 7 6 5 4 3 2 1 x 0 –6 –5 –4 –3 –2 –1 1 2 3 4 –1 Find the three inequalities which define the shaded triangle in the diagram. Answer [5]

5 marks

Mark scheme: 22 y [ 1, x Y 3, y Y x + 5 oe 5 B1 y R 1 B1 x R 3 B2 y R x + 5 or B1 y R – x + 5 where R is any inequality B1 all 3 inequalities correct

This question in 0580/23 Oct/Nov 2010

14 For y Examiner's Use 7 6 5 R 4 3 2 1 x 0 1 2 3 4 5 6 7 The region R is bounded by three lines. Write down the three inequalities which define the region R. Answer [4]

4 marks

Mark scheme: 14 y ≤ 5 4 B1 each inequality but accept any of the four x ≥ 2 inequality symbols y ≥ x Final B1 all 3 symbols correct

This question in 0580/22 Oct/Nov 2011

6 x is a positive integer and 15x – 43 < 5x + 2 . Work out the possible values of x. Answer [3]

3 marks

Mark scheme: 6 1, 2, 3, 4 3 M1 10x < 45 A1 x < 4.5

This question in 0580/22 May/June 2012

4 Solve the inequality. For 3y + 7 Y 2 – y Examiner's Use Answer [2]

2 marks

Mark scheme: 4 y ø –1.25 2 M1 inequality with y’s and constants correctly collected

This question in 0580/23 May/June 2012

9 Solve the inequality. 2 x − 3 x − Y=2 5 3 Answer [3]

3 marks

Mark scheme: 9 x Y 39 www 3 M1 correct first move M1 correct 2nd move M1 correct move to answer line

This question in 0580/21 Oct/Nov 2012

8 Solve the inequality. For Examiner′s 3x – 1 Ğ 11x + 2 Use Answer … [2] _____________________________________________________________________________________

2 marks

Mark scheme: 8 x ≥−3 oe 2 M1 for –3 ≤ 8x oe 8 If 0 then SC1 for −3 with incorrect inequality. 8

This question in 0580/21 May/June 2013

18 Solve 6x + 3 < x < 3x + 9 for integer values of x. Answer … [4] _____________________________________________________________________________________

4 marks

Mark scheme: 18 −1 −2 −3 −4 4 B3 for x < − 3 5 and x > −5.4 oe or B2 for x < − 3 5 or x > −5.4 oe or B1 for 5x < − 3 or − 9 < 2x oe Or mark on answer line −1 oe

This question in 0580/22 May/June 2013

16 Solve the inequality. x x - 2 + < 5 2 3 Answer … [4] _____________________________________________________________________________________

4 marks

Mark scheme: 16 x <6.8 4 B3 for 6.8 with wrong inequality or equal as answer. Or M1 for first move completed correctly and M1 for second move completed correctly and M1 for third move completed correctly  11 5 

This question in 0580/22 Oct/Nov 2013

15 Solve the inequality for positive integer values of x. 21 + x > x + 1 5 Answer … [4] __________________________________________________________________________________________ 1

4 marks

Mark scheme: 15 [0], 1, 2, 3 4 M1 for moving the 5 correctly M1 for collecting their terms A1 for a correct inequality for x eg [0 ≤ ] x < 4 12

This question in 0580/21 May/June 2014

4 Solve the inequality. 6n + 3  8n … [2]

2 marks

Mark scheme: 4 n < 1.5 oe final answer 2 B1 for 1.5 oe in answer or M1 for 3 > 8n − 6n oe 5 2 12 4

This question in 0580/22 Feb/March 2016

19 y 4 3 2 1 x –3 –2 –1 0 1 2 3 4 –1 Find the four inequalities that define the region that is not shaded. … … … … [5]

5 marks

Mark scheme: 19 y < 2 oe and x ⩾ –2 oe 2 B1 for either correct 1 1 y ⩾ x + 1 oe and y ⩽ – x + 3 oe 3 B2 for either y ⩾ x + 1 oe or y ⩽ – x + 3 oe 2 2 1 or SC2 for y = x + 1 oe and y = – x + 3 oe 2 1 or SC1 for y = x + 1 oe or y = – x + 3 oe 2 1 or SC4 for y ⩽ 2 oe , x > –2 oe, y > x + 1 oe 2 and y < – x + 3 oe

This question in 0580/22 Feb/March 2016

13 Solve the inequality. n + 7 1 5n – 8 … [2]

2 marks

Mark scheme: 13 n > 3.75 2 M1 for 7 + 8 < 5n – n oe

This question in 0580/21 May/June 2016

20 y 5 4 R 3 2 1 x 0 1 2 3 4 5 Find four inequalities that define the region, R, on the grid. … … … … [4]

4 marks

Mark scheme: 20 y < 4 4 B1 for each correct answer to a maximum of y ⩾ 3 3 marks. x ⩾ 2 First two may be combined as a single inequality y > x e.g. 3 ⩽ y < 4 for B2 After 0 scored SC1 for use of = signs or incorrect inequality signs in all four equations 9 6 + 4 8

This question in 0580/23 May/June 2016

21 y 4 NOT TO SCALE R x 0 1 4 Write down the three inequalities that define the unshaded region, R. … … … [4]

4 marks

Mark scheme: 21 y ⩾ 0 and x ⩾ 1 oe 4 SC3 for y > 0, x > 1 and x + y < 4 oe and or x + y ⩽4 oe B1 for y ⩾ 0 B1 for x ⩾ 1 oe and B2 for x + y ⩽ 4 oe or M1 for grad = –1 soi If B0 scored for first two B marks, SC1 for y = 0 and x = 1 or with incorrect inequality sign

This question in 0580/21 Oct/Nov 2016

7 Find the positive integers that satisfy the inequality t + 2 2 3 t - 6 . … [3]

3 marks

Mark scheme: 7 1, 2, 3 3 B2 for t < 4 or M1 for 2 + 6 > 3t – t oe or better If zero scored, SC1 for answer 0, 1, 2, 3 or 1, 2, 3, 4

This question in 0580/22 Oct/Nov 2016

24 y 7 6 5 4 R 3 2 1 0 x 0 1 2 3 4 5 6 7 8 9 10 11 Find the three inequalities that define the unshaded region, R. … … … [5]

5 marks

Mark scheme: 3 3 24 y ⩽ − x + 6 oe 5 SC4 for y < − x + 6, x > 2, y ⩾ x oe 5 5 x ⩾ 2 oe or y > x oe 3 B3 for y ⩽ − x + 6 oe 5 final answers 3 or B2 for y = − x + 6 oe 5 3 or B1 for gradient = – oe soi 5 and B2 for x ⩾ 2 and y > x oe or B1 for either x ⩾ 2 or y > x oe or for x = 2 and y = x with incorrect inequalities

This question in 0580/23 Oct/Nov 2016

10 y 5 4 3 2 1 x 0 1 2 3 4 5 6 –1 By shading the unwanted regions of the grid, find and label the region R that satisfies the following four inequalities. y G 2 y H 1 y G 2x - 1 y G 5 - x [3]

3 marks

Mark scheme: 10 Correct regionn identified 3 0 1 1 2 2 3 2 1 2 1 RR SC1 foor

This question in 0580/21 May/June 2017

13 Solve the inequality. 3n - 11 2 5n - 18 … [2]

2 marks

Mark scheme: 13 n < 3.5 oe final answer 2 M1 for 18 – 11 > 5n – 3n oe

This question in 0580/22 May/June 2017

16 (a) Solve the inequality. x + 13 H 3x + 7 … [2] (b) List the positive integers that satisfy the inequality in part (a). … [1]

3 marks

Mark scheme: 16(a) x ⩽ 3 final answer 2 M1 for 13 − 7 ⩾ 3x − x oe 16(b) 1, 2, 3 1FT correct answer or FT their answer to (a)

This question in 0580/23 May/June 2017

23 In one week, Neha spends x hours cooking and y hours cleaning. The time she spends cleaning is at least equal to the time she spends cooking. This can be written as y H x. She spends no more than 16 hours in total cooking and cleaning. She spends at least 4 hours cooking. (a) Write down two more inequalities in x and/or y to show this information. … … [2] (b) Complete the diagram to show the three inequalities. Shade the unwanted regions. y y = x 18 16 14 12 Hours 10 cleaning 8 6 4 2 x 0 2 4 6 8 10 12 14 16 18 Hours cooking [3] (c) Neha receives $10 for each hour she spends cooking and $8 for each hour she spends cleaning. Work out the largest amount she could receive. $ … [2]

7 marks

Mark scheme: 23(a) x + y ⩽ 16 oe 2 B1 for each mark final answers x ⩾ 4 oe If zero scored, SC1 for x + y < 16 and x > 4 23(b) Correct shading 3 M2 for lines at x = 4 and x + y = 16 or for correct shading of x < 4 or x + y > 16 or M1 for line at x = 4 or their x = 4 or for line at x + y = 16 or their x + y = 16 23(c) 144 2 M1 for (8, 8) selected or for 10 × x + 8 × y for any numerical point which is inside or on the boundary of their unshaded region

This question in 0580/21 Oct/Nov 2017

14 Find the integers which satisfy the inequality. - 5 1 2n - 1 G 5 … [3]

3 marks

Mark scheme: 14 –1, 0, 1, 2, 3 3 B2 for −<2 n - 3 or list with one error or omission or M1 for –5 + 1 < 2n or 2n ⩽ 5 + 1 or a list with 3 correct and no more than 1 incorrect or if zero scored, SC1 for 5, 3, 1, –1, –3

This question in 0580/22 Oct/Nov 2017

12 Solve the inequality. 3n - 5 2 17 + 8n … [2]

2 marks

Mark scheme: 12 2 2 M1 for 8n − 3n < −−5 17 or better n < − 4.4 or n < − 4 5 or 3n – 8n > 17 + 5 or better final answer

This question in 0580/21 May/June 2018

21 y 3 2 R 1 x 0 1 2 3 4 5 6 There are four inequalities that define the region R. One of these is y G x + 1. Find the other three inequalities. … … … [4]

4 marks

Mark scheme: 21 y ⩾ 1.5 oe 4 3 SC3 for y >1.5 oe and y > x oe and 4 y ⩾3 x oe 4 1 y ⩽ − x + 3 oe 1 2 y < − x + 3 oe 2 or B3 for any two correct inequalities or B1 for y ⩾ 1.5 oe and 3 1 B2 for y ⩾ x oe or y < − x + 3 oe 4 2 3 1 or y = x oe and y = − x + 3 oe or 4 2 with incorrect inequality signs 3 or B1 for y = x oe OR 4 1 y = − x + 3 oe or with incorrect 2 inequality signs

This question in 0580/21 May/June 2018

19 y 6 5 4 3 2 1 x 0 –2 –1 1 2 3 4 5 6 –1 –2 Find the two inequalities that define the region on the grid that is not shaded. … … [3]

3 marks

Mark scheme: 19 y > 2 oe final answer 3 B1 for y > 2 oe final answer y ⩾ 3 – x oe final answer B2 for y ⩾ 3 – x oe final answer or B1 for y = 3 – x oe soi or SC2 for y ⩾ 2 oe and y > 3 – x oe final answer

This question in 0580/22 May/June 2018

12 Find the integer values of n that satisfy the inequality 15 G 4n 1 28 . … [3]

3 marks

Mark scheme: 12 4, 5, 6 3 B2 for 1 error or 1 omission or M2 for 3.75 ⩽ n < 7 oe or M1 for 3.75 ⩽ n or n < 7 or better

This question in 0580/21 Oct/Nov 2018

6 Solve. 7m - 2 H 19 … [2]

2 marks

Mark scheme: 6 m ⩾ 3 final answer 2 M1 for correct first step e.g. 7m ⩾ 19 + 2

This question in 0580/22 Oct/Nov 2018

14 y 8 7 6 5 4 3 2 1 0 1 2 3 4 x By shading the unwanted regions of the grid, find and label the region R that satisfies the following four inequalities. x G 3 x H 2 y G 2x + 1 y H 4 - x [3]

3 marks

Mark scheme: 14 Correct reggion identifieed 3 B marks or SC1 for 1 2 1 2 3 2 R 0 1 2 1

This question in 0580/22 Oct/Nov 2018

16 y 10 8 6 4 2 –4 –2 0 2 4 6 8 10 12 14 x –2 –4 –6 + 6 y H 3x - 4 x + y H 5 y G- 12 x (a) By shading the unwanted regions of the grid, find and label the region R that satisfies the three inequalities. [2] (b) Find the largest value of x + y in the region R, where x and y are integers. … [1]

3 marks

Mark scheme: 16(a) R identified correctly 2 B marks 0 1 1 2 0 1 1 1 0 0 1 1 1 0 0 16(b) 7 1

This question in 0580/22 May/June 2019

9 Solve the inequality. x - 13 2 12 + 3x 2 … [2]

2 marks

Mark scheme: 9 x < –10 final answer 2 x M1 for –12 –13 > 3x – oe 2

This question in 0580/22 Oct/Nov 2019

16 y 6 5 4 3 2x + y = 6 2 1 0 1 2 3 4 5 x By shading the unwanted regions of the grid, find and label the region R that satisfies the following inequalities. y G 5 2x + y H 6 y H x + 1 [4]

4 marks

Mark scheme: 16 y = 5 ruled 4 B2 for two correct lines y = x + 1 ruled or B1 for one correct line Correct region indicated B2 for indication of correct region or B1 for shading that satisfies two of the inequalities

This question in 0580/22 Oct/Nov 2019

17 y 7 6 5 4 3 2 1 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 x – 1 – 2 – 3 – 4 – 5 By shading the unwanted regions on the grid, draw and label the region R that satisfies the following inequalities. - 2 1 x G 3 y G x + 3 [4]

4 marks

Mark scheme: 17 4 B1 for x = –2 dashed ruled line and x = 3 solid ruled line B1 for y = x + 3 solid ruled line B2 for indication of correct region or B1 for shading that satisfies two of the inequalities, e.g. two of x > –2, x ⩽ 3 and y ⩽ x + 3

This question in 0580/23 Oct/Nov 2019

13 y 4 3 2 R 1 x 0 1 2 3 4 Write down the three inequalities that define the region R. … … … [4]

4 marks

Mark scheme: 13 x + y < 4 4 B3 for any two correct or B1 for y ⩾ 1.5 y ⩾ 1.5 B2 for x + y < 4 or y ⩽ 2x + 1 or x + y = 4 and y = 2x + 1 or with incorrect y ⩽ 2x + 1 inequality signs or B1 for x + y = 4 or y = 2x + 1 or SC3 for > instead of ⩾ etc.

This question in 0580/23 May/June 2020

12 x is an integer and - 3 G 2 x - 1 1 3 . Find the values of x. … [2]

2 marks

Mark scheme: 12 –1, 0, 1 final answer 2 B1 for –1 ⩽ x < 2 or two correct answers and no extras or three correct answers and one extra/wrong

This question in 0580/23 May/June 2021

13 Solve. 6 - x 4 - 3 x H 5 … [3]

3 marks

Mark scheme: 13 x  1 final answer 3 6 M1 for 20 – 15x ⩾ 6 – x or 4 − 3x −x 5 5 M1 for correctly isolating terms in x FT their first step of dealing with the 5 x 6 20 – 6 ⩾ – x + 15x or −3x +  − 4 5 5

This question in 0580/21 Oct/Nov 2021

7 n – 2 – 1 0 1 Write down the inequality, in terms of n, shown by the number line. … [1]

1 marks

Mark scheme: 7 n > − 1 oe 1

This question in 0580/22 Feb/March 2022

15 Solve. 12x - 3 H 4x + 13 … [2]

2 marks

Mark scheme: 15 x ⩾ 2 final answer 2 M1 for 12x – 4x ≥ 13 + 3 oe

This question in 0580/23 May/June 2022

8 Solve. 30 (a) = 6 x x = … [1] (b) 11x - 3 H 2 ( 2x + 9) … [3]

4 marks

Mark scheme: 8(a) 5 1 8(b) x ⩾ 3 final answer 3 M1 for correct first step 11x – 3 ≥ 4x + 18 or 5.5x – 1.5 ⩾ 2x + 9 or better M1 for correctly collecting their x terms on one side and their number terms on the other side e.g. 11x – 4x ⩾ 18 + 3 or better

This question in 0580/22 May/June 2023

11 y 7 6 5 4 3 R 2 1 x – 1 0 1 2 3 4 5 6 7 – 1 Find the inequalities that define the unshaded region, R. … [4]

4 marks

Mark scheme: 11 y < x 4 B1 for y < x x < 6 B1 for x < 6 1 ⩽ y ⩽ 5 oe B2 for 1 ⩽ y ⩽ 5 or B1 for y ⩾ 1 or y ⩽ 5 If B0 scored, SC2 for y ⩽ x, x ⩽ 6 and 1 < y < 5 oe or SC1 for three correct from y = x, x = 6, y = 1 and y = 5

This question in 0580/21 May/June 2024

13 y 5 4 3 2 1 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 x – 1 – 2 – 3 – 4 – 5 The region R satisfies these inequalities. - 3 1 y G 2 y G x - 1 By drawing suitable straight lines and shading unwanted regions, find and label the region R. [4]

4 marks

Mark scheme: 13 4 B1 for y = 2 solid line B1 for y = x – 1 solid line B1 for y = –3 dashed line B1 for correct region identified satisfying the given inequalities R

This question in 0580/22 Oct/Nov 2024

7 x -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Write down the inequality for x represented on the number line. … [2]

2 marks

Mark scheme: 7 –3 < x ⩽ 4 2 B1 for –3< x or x ⩽ 4

This question in 0580/22 Feb/March 2025

16 y 6 5 4 R 3 2 1 0 1 2 3 4 x Write down all the inequalities that define the region R. … … … … [4]

4 marks

Mark scheme: 16 x ⩽ 2.5 4 B3 for answer y > 3 x < 2.5 y ⩾ 3 y < 4 y < 2x y ⩽ 4 y ⩽ 2x OR B1 for each If 0 or 1 scored, instead award SC2 for recognition of x = 2.5, y = 3, y = 4, y = 2x If 0 scored, SC1 for recognition of y = 2x

This question in 0580/21 Oct/Nov 2025

7 Represent the inequality - 4 1 x G 3 on the number line. x – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6 [2]

2 marks

Mark scheme: 7 2 B1 for open circle at –4 and closed circle at 3 or for a line between –4 and 3 (any circles)

This question in 0580/22 Oct/Nov 2025