Question

Simplify. $$|-18|$$

Short answer

Answer: The absolute value of -18 is 18.

Step-by-step solution

Understand the absolute value function

The absolute value function takes a number as input and returns the distance of that number from 0 on the number line. This is represented as: $$|x| = \begin{cases} x & \text{if}\ x \geq 0 \\ -x & \text{if}\ x < 0 \end{cases}$$ So, the absolute value of a positive number or zero is the same number, and the absolute value of a negative number is the opposite of that number (i.e., its positive counterpart).

Find the absolute value of -18

Using the definition of absolute value from step 1, we can find the absolute value of -18 by considering if it is greater than or equal to 0 or not. Since -18 is a negative number, we use the 2nd case of the definition we provided: $$|-18| = -(-18)$$

Compute the result

Now, we just need to compute the result: $$|-18| = -(-18) = 18$$ So, the absolute value of -18 is 18.

Key concepts

Negative Numbers

Negative numbers are less than zero. They are found to the left of zero on the number line. These numbers include things like

• -1, -2, -3,
• all the way to negative infinity.

Negative numbers are often used to represent things like debts, losses, or temperatures below freezing. When you take the absolute value of a negative number, you are essentially finding the positive equivalent of its distance from 0. For example, for oe{-18}, its absolute value becomes 18, because 18 is 18 spaces away from 0 on the number line. This concept is crucial when solving absolute value problems because it helps us understand that negatives become positive when we're looking at distance.

Positive Numbers

Positive numbers are those greater than zero. On the number line, these numbers sit to the right of zero. They include

• 1, 2, 3,
• and go all the way to positive infinity.

Positive numbers are often used to express quantities that can be added, such as money in the bank or a gain in temperature. For the absolute value function, a positive number's absolute value is the number itself. For example, oe{|5| = 5}. In the context of absolute value, remember that if you start with a positive, it stays the same because the question "How far is 5 from 0?" is answered simply as 5, as it is already in positive form.

Number Line

A number line is a visual representation of numbers positioned along a straight line. It extends infinitely in both directions and is used to easily compare numbers.

• The center of the number line is typically marked as zero.
• To the left lie negative numbers,
• and to the right, positive numbers.

This tool is essential for understanding the concept of absolute value, which focuses on the distance a number has from zero. In terms of distance, both -18 and 18 are equally spaced from zero, as they are located 18 units away from it on the number line. By measuring how far a number is from zero, the concept of absolute value becomes more intuitive, as it simply disregards direction (positive or negative) and concerns itself solely with distance.