Problem 35 - Elementary Algebra

Question

Determine each of the values, \(-|-(-1)|\)

Step-by-Step Solution

Answer

Answer: \(-1\)

Step 1: Evaluate the absolute value function

To find the value of the expression \(|-(-1)|\), we first determine the absolute value of the quantity inside the brackets, which is \(-1\). The absolute value of a number is the distance from that number to zero on the number line, irrespective of its sign. Therefore, the absolute value of \(-1\) is \(1\). Now, our expression becomes \(-|1|\).

Step 2: Apply the negative sign

Now, we just need to apply the negative sign in front of the absolute value function. We get \(-(1)\).

Step 3: Result

Multiply \(-1\) by \(1\) to get the final result: \(-1\). The value of the given expression \(-|-(-1)|\) is \(-1\).

Key Concepts

Absolute Value Function

The absolute value function is a cornerstone in understanding how to deal with distance on a number line. Visually and conceptually, the absolute value of a number is its distance from zero on the number line. This distance is always expressed as a non-negative number. For example, the absolute value of -3, written as \(|-3|\text{, is }3\).

The function is especially handy in various branches of mathematics and its applications. In fact, it's a recurring theme in solving algebraic expressions that involve absolute value because it's important to grasp that absolute value strips a number of its sign. Thus, when dealing with an expression like \(-|-(-1)|\), the absolute value of -1 is 1 since it's one unit away from zero on the number line. After determining the absolute value, you then apply any additional operations, which in this case involves applying the negative sign in front of the absolute value function.

Number Line

The number line is a visual representation of numbers laid out in a straight line, where each number is equidistant from its neighbors. It is an essential tool in understanding the concept of absolute value. On this line, zero is the central point, numbers to the right are positive, and numbers to the left are negative.

Whenever you're engaged with the absolute value, the number line helps to conceptualize the process. For negative numbers, such as -1, it is easy to see by looking at the number line that -1 is 1 unit to the left of zero. Even when the absolute value is contained within an algebraic structure, picturing the number line can guide you to the correct solution. It's this visual that assists students in confirming that the absolute value of any number, positive or negative, is its numerical value without regard to sign.

Negative Numbers

Negative numbers play a vital role in the concept of absolute values as they demonstrate how absolute value functions as a 'distance' rather than 'direction' measure. A negative number can be thought of as a certain quantity or value in the opposite direction from the positive side of the number line.

Understanding the role of negative numbers is crucial in algebra when you're required to navigate equations or expressions that involve calculating distances and differences. When an absolute value sign encases a negative number, such as in \(-|-(-1)|\), it's important to remember that the absolute value is addressing the magnitude only. Therefore, the negative sign in front of the absolute value symbol in our example will ensure that the final result remains negative.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a specific value when solved. These expressions can be simple or complex and may include absolute values, which require special attention.

The steps to simplify an algebraic expression that includes an absolute value, such as \(-|-(-1)|\), are methodical. First, you evaluate the absolute value, then apply any other operations prescribed by the expression. Through algebra, we see how different mathematical concepts like absolute value, negative numbers, and the number line can interact with each other to produce a coherent solution to problems, fostering a deeper understanding of how to manipulate and work with different types of numbers.