Problem 68 - Prealgebra

Question

Simplify each of the following. $$-|-5|$$

Step-by-Step Solution

Answer

The simplified form is -5.

Step 1: Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering the direction. It is always non-negative. So, \[ |-5| = 5 \]This is because you take the positive of the number inside the absolute value bars.

Step 2: Apply Negative Sign

Once you have found the absolute value, which is 5, apply the negative sign outside the absolute value expression. \[ -|-5| = -5 \]The negative sign in front of the absolute value means to take the negative of the resulting positive number.

Key Concepts

Simplifying Expressions

When simplifying expressions, especially those that involve absolute values and negatives, it's crucial to understand how to deal with each part of the expression correctly.
In this exercise, we started with the expression \(-|-5|\).
First, you must interpret the absolute value, which is indicated by vertical bars, like in \(|-5|\). The goal of the expression is to simplify it by removing these bars and considering the distance from zero, as absolute values always result in a non-negative number.
Once the absolute value is found (which is \(|-5| = 5\)), the next step is to handle any additional operations, like the negative sign in this expression. The negative sign outside the absolute value indicates that you make the final simplified result negative.
Thus, \(-|-5| = -5\). This step-by-step approach helps one deal with the components of the expression systematically to reach the correct simplified form.

Number Line

The number line is a fundamental tool for understanding concepts like absolute value and negative numbers. It visually represents numbers as points along a line, where the center point is zero.
Numbers to the right of zero are positive, while those to the left are negative. Each number's distance from zero can be interpreted as its absolute value, regardless of direction.
For instance, both +5 and -5 are located 5 units away from zero on the number line, hence their absolute values are the same: 5.
When simplifying expressions involving absolute values, think of the number line as a visualization technique that helps remove any confusion over signs and distances. It reminds us that the absolute value is strictly about magnitude without negativity. This means the absolute value of \(-5\) is \(+5\), as it only reflects the distance from zero.

Negative Numbers

Negative numbers are those values located to the left of zero on the number line. They have a "-" sign before a number and are used to indicate direction or loss, such as in debts or temperatures below zero.
Understanding negative numbers is vital when simplifying expressions because they often interact with absolute values.
A common point of confusion is how a negative sign outside an absolute value affects simplification. For example, \(-|-5|\) might look complicated but breaks down into a few understandable steps:

• First, calculate the absolute value, which turns the inner negative to positive (\(|-5| = 5\)).
• Then, apply the negative sign outside, which few students realize flips the positive result of the absolute value back to negative.

This results in \(-5\), showing how understanding negative numbers helps in achieving clarity and accuracy when simplifying complex expressions.