Rewrite each expression without absolute value bars. ||-3|-|-7||

Verified step by step guidance

1. Start by evaluating the innermost absolute value expressions. Recall that the absolute value of a number is its distance from zero, always non-negative. So, calculate \(| -3 |\) and \(| -7 |\) separately.
2. Replace \(| -3 |\) with its value and \(| -7 |\) with its value, simplifying the expression inside the outer absolute value bars.
3. Now, rewrite the expression as \(| \text{value of } | -3 | - \text{value of } | -7 | |\) without the absolute value bars inside, but keep the outer absolute value bars for now.
4. Calculate the difference inside the outer absolute value bars, then express the entire expression as the absolute value of that difference.
5. Finally, rewrite the expression without any absolute value bars by considering the definition of absolute value: for any real number \(x\), \(|x| = x\) if \(x \geq 0\), and \(|x| = -x\) if \(x < 0\). Use this to express the final result without absolute value bars.

Key Concepts

Absolute Value Definition

The absolute value of a number is its distance from zero on the number line, always expressed as a non-negative value. For example, |−3| equals 3 because −3 is three units from zero.

Evaluating Nested Absolute Values

When absolute value symbols are nested, evaluate from the innermost absolute value outward. Calculate the inner absolute value first, then apply the outer absolute value to the result.

Simplifying Expressions Without Absolute Value Bars

To rewrite expressions without absolute value bars, replace each absolute value with its non-negative equivalent. This often involves considering the sign of the inner expression and simplifying accordingly.