It's a question that pops up in math class, often as a quick check of understanding: if x is less than zero, what is the absolute value of x? It sounds simple, and in many ways, it is, but it hinges on a fundamental concept – the absolute value.
Think of absolute value as distance. It's how far a number is from zero on the number line, regardless of direction. So, the absolute value of 5 is 5, because it's 5 units away from zero. Similarly, the absolute value of -5 is also 5, because it's still 5 units away from zero, just in the opposite direction.
Now, let's bring in our condition: x < 0. This means x is a negative number. When we talk about the absolute value of a negative number, say -3, we're asking for its distance from zero. That distance is 3. How do we get from -3 to 3 mathematically? We multiply it by -1. So, if x is negative, its absolute value, |x|, is equal to -x.
Let's look at the options provided in a typical math problem. If we have x < 0, and we're asked for |x|:
- A. x²: If x is negative, x² will be positive (e.g., (-3)² = 9). This isn't always equal to |x| (e.g., | -3 | = 3, not 9).
- B. x: If x is negative, x is negative. The absolute value must be non-negative. So, this is incorrect.
- C. x - 2x: This simplifies to -x. As we discussed, if x is negative, -x is positive and represents the distance from zero. This fits!
- D. -2x: If x is negative, -2x will be positive, but it's twice the absolute value (e.g., if x = -3, -2x = 6, while |x| = 3).
So, when x is less than zero, the absolute value of x, |x|, is equal to -x. It's a neat way to ensure we're always talking about a non-negative quantity, representing that pure distance from the origin on the number line.