Question

True or False \(|x| \geq 0\) for any real number \(x\).

Short answer

True

Step-by-step solution

- Understand the Absolute Value

The absolute value of a number, denoted as \(|x|\), represents its distance from 0 on the number line, regardless of direction. By definition, absolute value is always non-negative.

- Analyze the Inequality

The problem asks whether \(|x| \geq 0\) is true for any real number \ x\. Given the definition of absolute value, no matter what real number \ x\ is (negative, positive, or zero), its absolute value cannot be less than zero, as distance is always zero or positive.

- Conclusion

Since the absolute value of any real number is always non-negative and hence equal to or greater than 0, the statement \(|x| \geq 0\) is indeed true for any real number \ x\.

Key concepts

absolute value

The absolute value of a number, denoted as \(|x|\), represents its distance from 0 on the number line. Whether a number is positive, negative, or zero, its distance from zero is always a non-negative value. For example:
\(|-5| = 5\)
\(|5| = 5\)
\(|0| = 0\).
Note that the absolute value ignores the direction on the number line. Therefore, while \(-5\) and \(5\) are on opposite sides of zero, they both have the same absolute value because their distance to zero is the same.

inequality

An inequality is a mathematical statement that compares two values. In this case, the exercise is asking about the inequality \(|x| \geq 0\). This specifies that the absolute value of \(x\) should be greater than or equal to zero. Here's why this is important:

• Equal sign (\(=\)): This part tells us that \(|x|\) can be exactly zero.
• Greater than sign (\(>\)): This indicates that the absolute value can also be any positive number.

Because absolute values are always non-negative, this inequality holds true for any real number.

non-negative numbers

Non-negative numbers are numbers that are either positive or zero; they are not negative. Examples include \(0\), \(3\), and \(12\). The absolute value of any real number falls into this category. This is because absolute value measures how far a number is from zero on the number line without considering direction. Therefore, the absolute value of a negative number such as \(-7\) becomes positive, which is also a non-negative number. In simpler terms, absolute value practically converts any number to its non-negative form.

real numbers

Real numbers include all the numbers you can think of along the number line. This means they include:

• Positive numbers: Values greater than zero
• Negative numbers: Values less than zero
• Zero: The neutral middle point
• Fractions and decimals: Such as \(1/2\) or \(2.75\)

The statement \(|x| \geq 0\) applies to all these types of numbers. Since real numbers encompass every possible value on the number line, and knowing that absolute value always results in a non-negative number, the inequality always holds true for any real number \(x\).