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Chapter 3: Problem 37

Determine each of the values, \(-(-|4|)\)

Short Answer

Expert verified
Answer: The value of the expression \(-(-|4|)\) is \(4\).

Step by step solution

01

Calculate the Absolute Value

We are given the absolute value of \(4\), as \(|-4|\). The absolute value is the distance from a number to \(0\) on the number line. So, for positive and negative numbers, their absolute value is the same positive number. Therefore, \(|-4| = 4\).
02

Simplify the Negation of the Absolute Value

Now we have \(-(4)\). A negative sign in front of a number means that we negate the number. So, we change the sign of \(4\) to its opposite. Therefore, \(-(4) = -4\).
03

Simplify the Double Negation

Finally, we have \(-(-4)\). This is the same as the negation of the result from step 2. When we see two negations in a row, they cancel each other out. So, the expression becomes \(-(-4) = 4\). The value of the expression \(-(-|4|)\) is \(4\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Expressions
Simplifying expressions is akin to cleaning up a cluttered room -- we aim to make it neat and understandable.
In mathematics, this process involves combining like terms, using the distributive property, and performing basic arithmetic to make expressions as compact and easy-to-use as possible.
In the case of the absolute value operation, it streamlines to a positive equivalent of a number, as absolute value reflects a number鈥檚 distance from zero regardless of its original sign.
So, simplifying the expression \( -(-|4|) \) starts with evaluating \( |4| \) to get \( 4 \), which is then followed by successfully applying negations.
Negation
Picture negation as a 'no' or 'reversal' sign.
When you place a minus before any value, you're essentially commanding it to 'do a U-turn.' In simpler terms, negation changes the sign of a number 鈥� positive values become negative and vice versa.
Using our exercise, \( -(4) \) flips the \( 4 \) into \( -4 \), as if saying 'no' to its positivity. Learning to recognize and apply negation is fundamental for understanding subtraction and solving equations in algebra.
Double Negation
Double negation in math acts much like a double negative in language; two 'nos' make a 'yes.'
Whhen confronted with two negative signs back-to-back, like in \( -(-4) \), they cancel each other out, reverting the number back to its positive form.
It's crucial to grasp this concept since it is used to simplify expressions and solve equations. In our exercise, the double negation turns \( -(-4) \) into \( 4 \) smoothly and swiftly.
Algebraic Operations
Fundamentally, algebraic operations are the building blocks of algebra -- adding, subtracting, multiplying, and dividing numbers, sometimes with unknowns (variables).
In the context of our exercise, we encounter two such operations: absolute value and negation.
The execution of these operations in the right order 鈥� first taking the absolute value and then applying the negations 鈥� productively leads us to the correct outcome. Every algebraic operation has a set of rules that, when followed, can simplify even the most tangled expressions.

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