Problem 34 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

Answer: -4

Step 1: Understand the absolute value of a negative number

The absolute value of a negative number is positive because it represents the distance between the number and zero on the number line, which is always nonnegative. In our case, we have \(|-(-4)|\).

Step 2: Find the absolute value of -4

To find the absolute value of -4, we simply remove the negative sign, as \(|-(-4)| = |4| = 4\).

Step 3: Solve the expression

Now that we have found the absolute value of -4, we can plug it into the original expression: \(-|-(-4)| = -|4| = -4\).

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They appear on the left side of the zero on a number line.
They are represented with a minus sign (-). For example, -1, -2, and -4 are all negative numbers.
These numbers have intriguing properties that can be confusing at first, but become easier to manage with practice. When dealing with negative numbers, remember that:

• Two negative signs make a positive. For instance, -(-4) is equal to 4.
• Negative numbers are always less than zero, meaning -4 is less than 0, but -4 plus 4 equals zero.

Understanding negative numbers is crucial, especially when performing calculations involving their absolute values or orders with positive integers.

Number Line

A number line is a useful tool for visualizing numbers and their relative positions. It's a straight line that shows numbers placed at equal intervals along its length. Zero is typically at the center.
Numbers to the right of zero are positive, and numbers to the left are negative. Here's why number lines are important:

• They help in understanding the concept of magnitude and comparisons between numbers.
• They allow us to easily see the "distance" of numbers from zero, which is essential for understanding absolute value.
• They are great for visualizing addition and subtraction, especially when negative numbers are involved.

Using a number line can simplify complex expressions by visualizing the arithmetic processes.

Expressions

Expressions in mathematics consist of numbers, variables, and operations like addition, subtraction, or absolute values. They are like sentences in math where we perform operations to simplify or find values.
In the exercise, \(-|-(-4)|\) is an expression.A closer look at expressions shows:

• Operations inside absolute value symbols, like \(|-(-4)|\), need to be resolved first.
• The order of operations (parentheses, exponents, multiplication and division, addition and subtraction) helps determine the sequence of evaluating an expression.

Understanding expressions is key for solving any math problem, as it enables you to break down each step systematically and logically.

Distance from Zero

The absolute value is a measure of a number's distance from zero on a number line, without considering direction.
For any number, positive or negative, its absolute value is always nonnegative.Here's what this means:

• For positive numbers and zero, the absolute value is the number itself.
• For negative numbers, the absolute value is the number without its negative sign. So, the absolute value of -4 is 4, as it is 4 units away from zero.

This concept helps in simplifying expressions, as seen in the exercise where \(|-(-4)| = |4| = 4\), showing how absolute values reduce complex problems into simpler arithmetic.