Problem 25 - Elementary Algebra

Question

Determine each of the values, \(-|-14|\)

Step-by-Step Solution

Answer

Answer: The value of the expression \(-|-14|\) is -14.

Step 1: Find the absolute value of -14

To find the absolute value of -14, represented as \(|-14|\), remember that the absolute value of a number is its distance from 0 on the number line. This means that if the number is negative, you should change its sign to make it positive. Therefore, the absolute value of -14 is 14. $$|-14| = 14$$

Step 2: Negate the absolute value

Now that we know \(|-14|=14\), we need to negate it as per the given expression \(-|-14|\). To do this, simply put a negative sign before the absolute value: $$-|-14| = -14$$ So, the value of the given expression is -14.

Key Concepts

Number Line

A number line is a visual representation of numbers along a straight line. It helps us understand the order and spacing of numbers, including positive and negative ones. On a number line:

• Zero is placed at the center.
• Positive numbers are found to the right of zero.
• Negative numbers are positioned to the left of zero.

This tool simplifies locating numbers and performing operations such as addition or subtraction. The position of a number on the line reveals its value and relationship to other numbers. For instance, -14 is found 14 units to the left of zero, making it a negative number. A number line aids in visualizing concepts such as distance and direction, essential for understanding operations involving integers like calculating absolute value.

Negative Numbers

Negative numbers are values less than zero, identified by a minus sign. They represent positions on a number line to the left of zero. In real-world contexts, negative numbers often signify:

• Temperature below freezing.
• Finances that are unpaid, like debt.
• Altitudes below sea level.

These numbers play a crucial role in mathematics when solving equations or finding the difference between values. In our exercise, -14 is a negative number, meaning it is 14 units away from zero in the opposite direction of positive numbers.

Distance from Zero

The distance from zero, in mathematics, is known as the absolute value. It measures how far a number is from zero on a number line, regardless of direction. This concept ignores negative or positive signs, focusing solely on magnitude. For example, the absolute value of -14 is 14, expressed as \(|-14| = 14\). Important points about absolute value include:

• It is always non-negative.
• Represented by vertical bars, such as \(|x|\).
• Helps understand symmetrical properties of numbers around zero.

The absolute value is pivotal when dealing with expressions where the sign could change the outcome, like the exercise at hand. Here, negating the absolute value leads us back to -14, completing the computation.