Problem 117 - Elementary Algebra

Question

Simplify. $$ -|-(-3)| $$

Step-by-Step Solution

Answer

The simplified expression is \(-3\).

Step 1: Understand the Problem

We need to simplify the expression \(-|-(-3)|\). The expression involves nested absolute values and a negative sign outside the absolute value.

Step 2: Simplify the Inner Absolute Value

First, consider the innermost part of the expression: \(-3\). The absolute value of \(-3\) is \(3\). So, we have:\[|-(-3)| = |3| = 3\]

Step 3: Apply the Negative Sign Outside

Apply the negative sign to the result from Step 2:\[-|3| = -3\]

Step 4: Final Simplified Expression

The simplified value of the expression \(-|-(-3)|\) is \(-3\).

Key Concepts

Understanding Absolute Value

Absolute value is a concept that refers to the distance of a number from zero on the number line, without considering direction.
This means that both positive and negative numbers have a non-negative absolute value. For example:

• The absolute value of 5 is 5: \(|5| = 5\)
• The absolute value of -5 is also 5: \(|-5| = 5\)

Absolute value is always zero or positive and is indicated by two vertical bars surrounding the number. So, it's quite handy when you want to simplify expressions by focusing only on numerical value itself, ignoring whether it's negative or positive.

Working with Negative Numbers

Negative numbers can sometimes seem tricky. But, understanding them is crucial for simplifying expressions.Negative numbers are those which are less than zero, like -1, -2, and so on. When working with them, it's important to remember:

• A negative times a negative is a positive: \((-2) \times (-3) = 6\)
• A negative times a positive is a negative: \((-2) \times 3 = -6\)
• Subtracting a negative is the same as adding its positive: \(5 - (-2) = 5 + 2 = 7\)

These rules are essential to ensure that we handle negative signs correctly in mathematical expressions and computations.

Exploring Nested Operations

Nested operations refer to expressions that contain operations within operations, like \(3 - (4 - 2)\). Handling these requires carefully following the order of operations, often remembered by PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).In nested operations:

• You always tackle the innermost operation first. This might involve simplifying what's inside brackets or absolute value symbols.
• Once the inner computations are complete, move outward, step by step, to simplify the whole expression.
• Using absolute value in nested operations might mean applying the absolute value to an already simplified part before moving on.

For example, in the expression \(-|-(-3)|\), we first simplify the innermost part to find \(|-3|\), then apply the external negative.
The step-by-step approach ensures that the solution is both accurate and understandable.