Which one of the following is correct? (a) Statement 1 is true, and Statement 2 is false. (b) Statement 2 is true, and Statement 1 is false. (c) Both the statements are true. (d) B... Which one of the following is correct? (a) Statement 1 is true, and Statement 2 is false. (b) Statement 2 is true, and Statement 1 is false. (c) Both the statements are true. (d) Both the statements are false.
Understand the Problem
The question is asking to evaluate two mathematical statements about inequalities and determine which of the provided options correctly states the truth of these statements.
Answer
(a) Statement 1 is true, and Statement 2 is false.
Answer for screen readers
(a) Statement 1 is true, and Statement 2 is false.
Steps to Solve
- Evaluate Statement 1
We will solve the inequality (2x - 1 < 5).
Adding 1 to both sides:
$$ 2x < 6 $$
Dividing both sides by 2:
$$ x < 3 $$
Now, the solution for Statement 1 is (x < 3). The statement claims (x \in {1, 2}). Both values (1 and 2) satisfy (x < 3), so Statement 1 is true.
- Evaluate Statement 2
We will solve the inequality (|x - 2| < 3).
This absolute value inequality can be rewritten as two separate inequalities:
$$ -3 < x - 2 < 3 $$
Adding 2 to all parts:
$$ -1 < x < 5 $$
This implies that the solution set is (x \in (-1, 5)).
The statement claims that the solution set is ({x: x \leq 1 \text{ or } x \geq 5}), which does not match. Therefore, Statement 2 is false.
- Conclude which statements are true or false
Based on the evaluations:
- Statement 1 is true.
- Statement 2 is false.
Thus, the correct option is (a): Statement 1 is true, and Statement 2 is false.
(a) Statement 1 is true, and Statement 2 is false.
More Information
In this problem, understanding inequalities and absolute values is crucial. The evaluations of the statements highlight how to manipulate inequalities, and it’s essential to differentiate between strict and non-strict inequalities.
Tips
- Misinterpreting absolute value inequalities, such as forgetting to consider both cases when splitting the absolute value.
- Incorrectly deriving the boundaries of inequalities, especially when working with multiple conditions.
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