Final answer:
The true statements about the absolute value of a negative rational number x are that the absolute value of x is the opposite of x, |x| is a positive number, the absolute value of x is a positive number, |x| is greater than x, and the absolute value of x is greater than x.
Step-by-step explanation:
Let x be any negative rational number. Given this information, the following statements can be evaluated:
- |x| is the opposite of x - False, since |x| is the absolute value of x, which means it is a positive number, whereas x is negative.
- The absolute value of x is the opposite of x - True, because the absolute value of a negative number is its positive counterpart.
- |x| is equal to x - False, |x| cannot be equal to x because |x| is positive and x is negative.
- The absolute value of x is equal to x - False, for the same reason as above.
- |x| is a positive number - True, as the absolute value of any number is always non-negative.
- The absolute value of x is a positive number - True, this is the definition of absolute value.
- |x| is greater than x - True, because |x| is positive and x is negative.
- The absolute value of x is greater than x - True, for the same reason as above.
- |x| is greater than the distance from 0 to x on the number line - False, the absolute value of x represents that exact distance, so it cannot be greater than itself.
Therefore, the true statements about the absolute value of a negative rational number x are that the absolute value of x is the opposite of x, |x| is a positive number, the absolute value of x is a positive number, |x| is greater than x, and the absolute value of x is greater than x.