Problem 125 - Elementary Algebra

Question

Simplify. $$ -(-\operatorname{abs}(9)) $$

Step-by-Step Solution

Answer

The simplified expression is 9.

Step 1: Understand the Problem

The task is to simplify the expression \[-(-\operatorname{abs}(9))\].Here, \(\operatorname{abs}(9)\) denotes the absolute value of 9 and the negative signs need to be simplified.

Step 2: Calculate the Absolute Value

The absolute value function, \(\operatorname{abs}(x)\), returns the non-negative value of \(x\). Hence, \(\operatorname{abs}(9) = 9\).

Step 3: Resolve the Inner Negative

Substitute the calculated absolute value back into the expression to simplify: \(-(-9)\).

Step 4: Simplify Double Negation

The double negative \(-(-9)\) simplifies to positive \(9\), because the negatives cancel each other out.

Key Concepts

Absolute Value

The concept of _absolute value_ is central to solving many algebraic expressions.
When you encounter the absolute value, denoted as \( \operatorname{abs}(x) \), it refers to the distance between the number \( x \) and zero on the number line.
The key aspect to remember is that this value is always a non-negative number.
It essentially "strips away" any negative sign, leaving you with

• \( \operatorname{abs}(9) = 9 \)
• \( \operatorname{abs}(-9) = 9 \)
• \( \operatorname{abs}(0) = 0 \)

These examples illustrate how the absolute value reflects the magnitude of a number without concern for direction or sign. By understanding this concept, you can simplify expressions involving complex evaluations by isolating the absolute value first.

Double Negation

Double negation occurs when a number or expression is negated twice, such as \(-(-x)\).
In simpler terms, you can think of negation as "flipping" the sign. So when something is negated twice, these two actions cancel each other out.
For example, when you have \(-(-9)\), the two negatives effectively "flip" the sign back to its original state, yielding a positive number:

• Start with the inner negative: \(-9\)
• Apply the outer negative: \(-(-9) = 9\)

Understanding double negation helps you quickly simplify expressions where the sign needs to be systematically adjusted back to its positive or original form.
This is an essential concept when working with equations and simplifies the solution process significantly.

Algebraic Simplification

Algebraic simplification is the process of transforming an expression into its simplest form. This involves eliminating any unnecessary components or complex elements without changing the expression's value.
For instance, when simplifying \(-(-\operatorname{abs}(9))\), multiple steps are taken to reach the simple answer of \(9\):

• Evaluate the absolute value: \(\operatorname{abs}(9) = 9\)
• Resolve the double negative: \(-(-9) = 9\)

These steps show how each part of the expression can be streamlined to its simplest form, ensuring the computation remains accurate.
Simplification is crucial because it makes expressions easier to handle, reduces errors, and enhances understanding of the mathematical principles involved. By mastering simplification, you can solve problems more efficiently and with greater clarity.