What is the solution, if any, to the inequality ∣3x∣≥0?
The Correct Answer and Explanation is:
To solve the inequality (|3x| \geq 0), we need to analyze the absolute value expression (|3x|).
The absolute value of a number is always non-negative, meaning it can either be zero or positive. Therefore, (|3x| \geq 0) is always true for any real number (x). Let’s break this down further:
- Understanding Absolute Value: The expression (|3x|) represents the distance of (3x) from zero on the number line. Regardless of whether (3x) is positive, negative, or zero, the absolute value will either equal zero or a positive value.
- Finding the Zero Point: The only time (|3x| = 0) is when (3x = 0). This occurs when (x = 0). Thus, (|3x|) is equal to zero at this point.
- Positive Values: For all other values of (x) (whether positive or negative), (|3x|) is positive. For example:
- If (x > 0), then (3x > 0) and (|3x| = 3x > 0).
- If (x < 0), then (3x < 0) and (|3x| = -3x > 0).
- Conclusion: Since (|3x|) can only be zero or positive, it is always greater than or equal to zero for any real number (x). This leads us to conclude that the inequality (|3x| \geq 0) holds true for all real numbers.
- Final Solution: Therefore, the solution set to the inequality (|3x| \geq 0) is all real numbers. In interval notation, this is expressed as ( (-\infty, \infty) ).
In summary, because the absolute value of any expression is either zero or positive, and since the expression (|3x|) satisfies this condition for all values of (x), we conclude that the inequality (|3x| \geq 0) is valid for every real number (x).