Predicate:
1 ≤ x^2 ≤ 4,
domain: R
The Correct Answer and Explanation is:
The given predicate is:
[ 1 \leq x^2 \leq 4 ]
Domain: R (Real Numbers)
Interpretation:
The inequality expresses a condition on the value of ( x^2 ). The domain is the set of real numbers ( \mathbb{R} ), meaning that ( x ) can take any real value.
Breaking down the inequality:
- First part of the inequality: ( 1 \leq x^2 )
- This means that ( x^2 ) must be greater than or equal to 1.
- Since ( x^2 \geq 0 ) for all real ( x ) (because squaring any real number results in a non-negative number), this condition restricts ( x ) to values such that the square of ( x ) is at least 1. This can happen in two cases:
- ( x \geq 1 ) or
- ( x \leq -1 )
- Second part of the inequality: ( x^2 \leq 4 )
- This means that ( x^2 ) must be less than or equal to 4.
- For the square of ( x ) to be less than or equal to 4, ( x ) must lie between ( -2 ) and ( 2 ), inclusive. That is:
- ( -2 \leq x \leq 2 )
Combining the two inequalities:
- From the first condition (( 1 \leq x^2 )), we know that ( x ) must be outside the range ( (-1, 1) ), i.e., ( x \geq 1 ) or ( x \leq -1 ).
- From the second condition (( x^2 \leq 4 )), we know that ( x ) must lie within the range ( -2 \leq x \leq 2 ).
Thus, combining both conditions, we conclude that ( x ) must satisfy:
[ x \in [-2, -1] \cup [1, 2] ]
Final Answer:
The solution to the predicate ( 1 \leq x^2 \leq 4 ) in the domain of real numbers is:
[ x \in [-2, -1] \cup [1, 2] ]
This means that ( x ) can be any value in the intervals ( [-2, -1] ) or ( [1, 2] ).