In Exercises 51–60, rewrite each expression without absolute value bars. ||-3|-|-7||

Verified step by step guidance

1. Start by evaluating the innermost absolute value: |-7|. Since the absolute value of a number is its distance from zero on the number line, |-7| becomes 7.
2. Next, evaluate the absolute value of -3: |-3|. The absolute value of -3 is 3.
3. Now, substitute these values back into the expression: ||3|-7|.
4. Evaluate the expression inside the absolute value bars: |3 - 7|. This simplifies to |-4|.
5. Finally, evaluate the absolute value of -4: |-4|, which is 4.

Key Concepts

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|, and is always non-negative. For example, |-3| equals 3, as it measures the distance of -3 from 0.

Properties of Absolute Value

Absolute value has specific properties that are useful in simplifying expressions. One key property is that |a| - |b| can be rewritten as |a - b| when both a and b are non-negative. Understanding these properties helps in manipulating expressions involving absolute values effectively.

Nested Absolute Values

Nested absolute values occur when absolute value expressions are contained within other absolute value expressions. To simplify these, one must evaluate the innermost absolute value first, then apply the outer absolute value. This step-by-step approach is crucial for correctly rewriting complex expressions.