Question

Consider the following statements:
Statement-I: The set of all solutions of the linear inequalities 3x + 8 < 17 and 2x + 8 ≥ 12 are x < 3 and x ≥ 2 respectively.
Statement-II: The common set of solutions of the linear inequalities 3x + 8 < 17 and 2x + 8 ≥ 12 is (2,3).
Which of the following is true?

• A. Statement-I is false but Statement-II is true
• B. Both the statements are true
• C. Both the statements are false
• D. Statement-I is true but Statement-II is false

Hint:

When solving linear inequalities, always check the boundary conditions to ensure the solution set is valid.

Answer 1

The Correct Option is D

We need to evaluate the two statements given in the question: 
Statement-I:
We are given two linear inequalities: 3x + 8 < 17 and 2x + 8 ≥ 12. Let's solve each inequality. 
- Solve 3x + 8 < 17: 3x < 17 - 8 ⇒ 3x < 9 ⇒ x < 3. Thus, the solution to the first inequality is x < 3. 

- Solve 2x + 8 ≥ 12: 2x ≥ 12 - 8 ⇒ 2x ≥ 4 ⇒ x ≥ 2. Thus, the solution to the second inequality is x ≥ 2. 

Therefore, Statement-I is true since x < 3 and x ≥ 2 correctly represent the solution of the two inequalities. 

Statement-II:
We need to check the common solution set for both inequalities. 
From Statement-I, we know that: x < 3 and x ≥ 2. Thus, the common solution set should be x ∈ [2, 3), but Statement-II mentions x ∈ [2, 3], which is incorrect because x = 3 does not satisfy the inequality 3x + 8 < 17. 

Hence, Statement-II is false. Thus, the correct answer is that Statement-I is true but Statement-II is false.