IGCSE · Mathematics 0580 · Algebra & graphs

Inequalities: IGCSE Mathematics questions with answers

Free Inequalities questions for IGCSE Mathematics (0580), part of Algebra & graphs. Each question has the answer and the marking points an examiner would look for.

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Inequalities practice questions

Question 1Structured · 8 marksA* challenge

The region R is defined by the three inequalities
y ≥ x2 − 1, y ≤ 6, y > 2x.

(a)

Find the coordinates of the points where the line y = 6 meets the curve y = x2 − 1.

[2]
(b)

Find the number of points with integer coordinates that lie in R.

[4]
(c)

Find the greatest value of x + y for a point with integer coordinates in R.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)(−√7, 6) and (√7, 6)
x2 = 7 M1
x = ±√7 A1
2
(b)23
considers each integer x from −2 to 2 M1
correct range of y for at least three values of x M1
excludes points on y = 2x M1
23 A1
4
(c)8
tests candidate points on the boundary M1
8 A1
2
Question 2Structured · 4 marksA* challenge
(a)

Solve x2 − 7x + 10 ≤ 0.

[3]
(b)

Write down the number of integers that satisfy the inequality.

[1]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)2 ≤ x ≤ 5
(x − 2)(x − 5) M1
critical values 2 and 5 A1
2 ≤ x ≤ 5 A1
3
(b)4
4 B1
1
Question 3Structured · 3 marks
(a)

Solve 5x + 4 < 24.

[2]
(b)

List the positive integers that satisfy the inequality.

[1]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)x < 4
5x < 20 M1
x < 4 A1
2
(b)1, 2, 3
1, 2, 3 B1
1
Question 4Structured · 3 marks

Solve the inequality -5 < 2x + 1 ≤ 9 and list the integer values of x.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
-3 < x ≤ 4; −2, −1, 0, 1, 2, 3, 4
subtract 1 and divide by 2 M1
-3 < x ≤ 4 A1
integers listed B1
3
Question 5Structured · 4 marks
(a)

Solve 6x − 9 < −45.

[2]
(b)

List all the integers n that satisfy −1 < 2n + 1 ≤ 9.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)x < −6
6x < −36 M1
x < −6 A1
2
(b)0, 1, 2, 3, 4
-1 < n ≤ 4 seen M1
0, 1, 2, 3, 4 A1
2
Question 6Structured · 3 marksA* challenge

Find all the integer values of n that satisfy
−9 < 1 − 3n ≤ 10.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
−3, −2, −1, 0, 1, 2, 3
−3 ≤ n or n < 103 (sign reversed correctly) M1
−3 ≤ n < 103 A1
−3, −2, −1, 0, 1, 2, 3 A1
3

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Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in