Problem 58 - Beginning and Intermediate Algebra

Question

Find each absolute value. \(-\left|-\frac{4}{5}\right|\)

Step-by-Step Solution

Answer

-\(\frac{4}{5}\)

Step 1: Understand the Absolute Value

Absolute value of a number is the distance of that number from zero on the number line, regardless of the direction. It is always non-negative. The absolute value of a number is denoted by \(|\text{number}|\).

Step 2: Identify the Number Inside the Absolute Value

The given expression is \(-|-\frac{4}{5}|\). Identify the number inside the absolute value. Here, the number is \(-\frac{4}{5}\).

Step 3: Compute the Absolute Value

Calculate the absolute value of \(-\frac{4}{5}\). Since the absolute value of a number discards the negative sign, we have \(|-\frac{4}{5}| = \frac{4}{5}\).

Step 4: Apply the Negative Sign Outside the Absolute Value

Substitute the computed absolute value back into the original expression and include the negative sign outside the absolute value: \(-|-\frac{4}{5}| = -\frac{4}{5}\).

Key Concepts

distance on number line

Understanding the concept of distance on the number line is crucial when dealing with absolute values. The number line is a visual representation of numbers placed at equidistant points along a straight line. When we talk about the distance of a number from zero on a number line, we are essentially discussing how far the number is from zero, regardless of its direction.
This distance is always reported as a positive value because distance cannot be negative.

• For instance, the distance of 4 from zero is 4 units.
• Similarly, the distance of -4 from zero is still 4 units.

This equals to saying that both +4 and -4 are 4 units away from zero. Understanding this will help you evaluate absolute values with ease.

non-negative value

The term 'non-negative' means that a value cannot be negative. In the context of absolute value, this means that whenever you calculate the absolute value of a number, the result will always be a non-negative quantity.
Here's why:
The absolute value of a number is simply the distance it is from zero, and distance cannot be negative. It is always zero or positive.

• For instance, if the number is 5, its absolute value is |5| = 5.
• If the number is -3, its absolute value is |-3| = 3.

Even though -3 is negative, the absolute value makes it positive because it measures how far -3 is from zero, which is 3 units.

evaluate expressions

Evaluating expressions involving absolute values can be straightforward if you follow a few steps. Let's break down the solved exercise step by step to highlight these steps: Step 1: Understand the Absolute Value
The absolute value of a number represents its distance from zero on the number line, and it is always non-negative. It is denoted as \(|\text{number}|\).
Step 2: Identify the Number Inside the Absolute Value
In the exercise given, we need to find the absolute value of \(-|-\frac{4}{5}|\). The number inside the absolute value is \(-\frac{4}{5}\).\