IMO Practice Test: Linear Inequations | Class 10 Mathematics

Question 1 of 6 medium

Solve: 3(x − 1) − 2(x + 3) > 2x − 7

x < −2

x > −2

x < 2

x > 2

Explanation: Expand: $3(x-1)-2(x+3)=3x-3-2x-6=x-9$. So $x-9 > 2x-7 \Rightarrow -9+7 > 2x-x \Rightarrow -2 > x$, i.e. $x < -2$.

Question 2 of 6 medium

Find the greatest integer that satisfies: 3(2x − 5) < 5x + 2

15

16

17

18

Explanation: Solve $3(2x-5) < 5x+2$: $6x-15 < 5x+2 \Rightarrow x < 17$. The greatest integer satisfying $x < 17$ is $16$.

Question 3 of 6 medium

Solve: 2 ≤ (3x − 1)/2 < 5, x ∈ integers

{2,3}

{2,3,4}

{1,2,3}

{1,2,3,4}

Explanation: Multiply by $2$: $4 \le 3x-1 < 10$. Add $1$: $5 \le 3x < 11$. Divide by $3$: $\frac{5}{3} \le x < \frac{11}{3}$, i.e. $1.67 \le x < 3.67$. The integer solutions are $\{2,3\}$.

Question 4 of 6 medium

The solution set of |x − 2| > 5 is:

x < −3 or x > 7

−3 < x < 7

x < 7

x > −3

Explanation: $|x-2| > 5$ means $x-2 > 5$ or $x-2 < -5$. The first gives $x > 7$; the second gives $x < -3$. So the solution set is $x < -3$ or $x > 7$.

Question 5 of 6 medium

If x is an integer, how many integers satisfy 2x − 7 < x + 3 ≤ 3x − 5?

4

5

6

7

Explanation: From $2x-7 < x+3$ we get $x < 10$. From $x+3 \le 3x-5$ we get $8 \le 2x$, i.e. $x \ge 4$. So $4 \le x < 10$, giving the integers $4,5,6,7,8,9$ — that is $6$ integers.

Question 6 of 6 medium

Represent the solution of 4x − 7 ≥ 3x + 5 on a number line. The graph shows:

Closed circle at 12, arrow right

Open circle at 12, arrow right

Closed circle at 12, arrow left

Open circle at 12, arrow left

Explanation: 4x−3x ≥ 5+7 → x ≥ 12 → closed circle at 12, arrow right

IMO Practice Test — Linear Inequations