IMO Practice Test — Inequalities (One Variable)
6 Questions • 25 min • Olympiad level
25:00
Question 1 of 6
hard
If 3x - 7 > 2x + 5, which interval contains x?
x > 12
x < 12
x > -12
x < -12
Explanation: 3x - 2x > 5 + 7 => x > 12
Question 2 of 6
hard
Find the largest integer satisfying (2x-1)/3 <= 5.
6
7
8
9
Explanation: 2x-1 <= 15 => 2x <= 16 => x <= 8. Largest integer is 8.
Question 3 of 6
hard
Solve the compound inequality: 4 <= 2x - 6 < 10
5 <= x < 8
5 < x <= 8
4 <= x < 10
2 <= x < 5
Explanation: Add 6 throughout: 10 <= 2x < 16. Divide by 2: 5 <= x < 8.
Question 4 of 6
hard
A student needs an average of at least 80 on 4 tests. First 3 scores: 75, 82, 78. What is the minimum score on the 4th test?
80
83
85
88
Explanation: (75+82+78+x)/4 >= 80 => 235+x >= 320 => x >= 85
Question 5 of 6
hard
Solve: 5 - 3x > 2x + 20
x > -3
x < -3
x > 3
x < 3
Explanation: -3x - 2x > 20 - 5 => -5x > 15 => divide by -5, flip sign: x < -3
Question 6 of 6
hard
Solve: 5x + 3 > 2x + 12. Which integers satisfy this?
x > 2 (all integers > 2)
x > 3 (all integers > 3)
x < 3 (all integers < 3)
x < 2 (all integers < 2)
Explanation: 5x - 2x > 12 - 3 => 3x > 9 => x > 3. So all integers greater than 3.