Determine whether each statement is true or false. |8-12| = |8| - |12|

Verified step by step guidance

1. Recall the definition of absolute value: for any real number \(x\), \(|x|\) represents the distance of \(x\) from zero on the number line, and it is always non-negative.
2. Evaluate the left side of the equation: \(|8 - 12|\). First, calculate the expression inside the absolute value: \$8 - 12 = -4\(. Then, find the absolute value: \)|-4|$.
3. Evaluate the right side of the equation: \(|8| - |12|\). Calculate each absolute value separately: \(|8|\) and \(|12|\), then subtract the results.
4. Compare the two results from the left and right sides to determine if the equation \(|8 - 12| = |8| - |12|\) holds true.
5. Remember that the absolute value of a difference is not generally equal to the difference of absolute values, so consider this property when making your conclusion.

Key Concepts

Absolute Value Definition

The absolute value of a number represents its distance from zero on the number line, always yielding a non-negative result. For example, |8| = 8 and |-12| = 12, regardless of the sign of the original number.

Properties of Absolute Value

Absolute value has specific properties, such as |a - b| representing the distance between numbers a and b. However, absolute value does not distribute over subtraction, meaning |a - b| is not generally equal to |a| - |b|.

Evaluating True or False Statements

To determine if a statement involving absolute values is true or false, substitute values and apply definitions and properties. For example, calculate |8 - 12| and compare it to |8| - |12| to verify the equality.