When considering the absolute value of a number, we are looking at the distance of that number from zero on the number line, regardless of its sign. To determine whether the absolute value of x is greater than one, we need to compare the absolute value of x to one. Mathematically, we can write this as |x| > 1. Let’s explore this question further.
**YES**
If |x| > 1, then it means that the distance of x from zero on the number line is greater than one. In other words, x is situated more than one unit away from zero in either direction. This indicates that x could be any number greater than 1 or less than -1, making the absolute value of x greater than one.
FAQs:
1. What does |x| > 1 mean?
This inequality signifies that the absolute value of x is greater than one, implying that x is situated more than one unit away from zero on the number line.
2. Is the absolute value of a number always positive?
Yes, the absolute value of a number is always non-negative, as it represents the distance of that number from zero on the number line.
3. Can the absolute value of a number be negative?
No, the absolute value of a number cannot be negative because it denotes the distance of that number from zero, which is always a non-negative value.
4. How does the absolute value of x being greater than one affect its possible values?
If |x| > 1, it means that x can be any number greater than 1 or less than -1, as those are the values that are more than one unit away from zero.
5. What does it mean if |x| is less than 1?
If |x| < 1, it signifies that x is situated less than one unit away from zero on the number line, indicating that x could be any number between -1 and 1.
6. Is the absolute value of zero greater than one?
No, the absolute value of zero is exactly one, as it represents the distance of zero from zero on the number line.
7. Can the absolute value of a number be zero?
Yes, the absolute value of zero is zero, as it denotes the distance of zero from zero on the number line.
8. Does the absolute value of x being greater than one imply x is positive?
No, the absolute value of x being greater than one indicates that x is more than one unit away from zero, which could be a positive or negative number.
9. How is the absolute value of a number calculated?
To calculate the absolute value of a number x, you remove the negative sign if it exists. If x is positive or zero, then |x| = x. If x is negative, then |x| = -x.
10. What is the significance of the absolute value in mathematics?
The absolute value is crucial in determining distances, magnitudes, and comparisons in mathematics, as it provides a non-negative representation of a number’s value.
11. Can the absolute value of x be equal to one?
Yes, the absolute value of x can be equal to one, which occurs when x is either 1 or -1, as they are exactly one unit away from zero on the number line.
12. How does understanding absolute value help in problem-solving?
Understanding absolute value allows us to analyze distances, magnitudes, and relationships between numbers more effectively, aiding in solving various mathematical problems and equations.