Problem 68 - Algebra A Combined Function

Question

Simplify each of the following. See Example 17. $$ -|-5| $$

Step-by-Step Solution

Answer

The simplified expression is -5.

Step 1: Understand the Absolute Value

Absolute value is the distance a number is from zero on the number line, regardless of direction. For any number \( x \), the absolute value is denoted as \( |x| \). This means \( |-5| \) equals 5 because it is 5 units away from zero.

Step 2: Apply Negation in Front of the Absolute Value

The exercise includes a negative sign in front of the absolute value expression: \(-|-5|\). Since \( |-5| = 5 \), substitute 5 into the expression which becomes \(-5\).

Step 3: Simplify the Expression

Now simplify the expression by applying the negation. Since you have \(-5\), the simplified expression is \(-5\).

Key Concepts

Understanding the Number Line

The number line is a simple visual representation of numbers placed in increasing order from left to right. It's an essential tool in mathematics for understanding the relationship and positioning of numbers, including their absolute values.
Imagine a straight line with zero at its center. To the right, you'll find positive numbers, and to the left, negative numbers. Each number indicates a specific point on this line.

• Absolute value: Measures the distance from zero, regardless of the side of the line the number is.
• For example, \(-5\) on the number line is five units away from zero, just like \(+5\).

To solve problems involving absolute values, visualize which point a number occupies on the number line. This will help you understand how far a number is from zero. It's logical how negative and positive signs appear in problem solving, particularly when dealing with abs<|vq_11267|>{

The Concept of Negation in Mathematics

Negation, in mathematics, refers to reversing the sign of a number. When you negate a positive number, it becomes negative and vice versa. This operation switches its direction on the number line.
Let's look at the exercise example of \(-|-5|\). Here are the steps to understand negation:

• Find the absolute value of \(-5\), which is \(+5\), as it's five steps away from zero.
• Apply negation by placing a negative sign in front of \(+5\), changing it to \-5\.

The operation of negating a number effectively moves it across the number line from one side of zero to the other. This is essential when simplifying expressions. It explains how expressions change their value through simple sign reversal.

Simplifying Mathematical Expressions

Simplifying expressions is a fundamental skill in mathematics. It involves reducing the expression to its most basic form. Simplification helps in solving problems more effectively and arriving at clearer answers.
For instance, let's simplify the expression \(-|-5|\):

• First, compute the absolute value of the inner expression, which is \|-5| = 5\.
• Next, address the entire expression \-|-5|\ by applying the negation, resulting in \-5\.

Always remember that simplification involves following the rules of operations in math carefully. Make sure to apply absolute values and negation properly. This guarantees your final expression is elegantly simplified, leaving no room for errors in calculation. Each piece of this process builds your confidence in handling more complex mathematical problems.