Problem 35 - Algebra 2

Question

Find each absolute value. $$ |-1.5| $$

Step-by-Step Solution

Answer

|-1.5| = 1.5

Step 1: Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative.

Step 2: Identify the Number

The given number is -1.5 . We need to find its absolute value.

Step 3: Apply Absolute Value Definition

Since absolute value represents distance which is always positive, the absolute value of -1.5 is obtained by removing the negative sign.

Step 4: Calculation

So, |-1.5| equals to positive 1.5 .

Key Concepts

Number Line

Visualizing numbers on a number line can be a helpful way to understand numerical concepts like absolute value. Imagine a horizontal line where numbers are marked at equal intervals. Zero is placed at the center, with positive numbers to the right and negative numbers to the left.

Here’s why a number line is crucial:

• It helps in visualizing the position of various numbers in relation to one another.
• It provides a clear view of how far a number is from zero, which directly relates to the concept of absolute value.

To mark \(-1.5\), locate it on the number line to the left of zero. By observing its position, you can easily see that the absolute value involves the distance it covers to reach zero, focusing just on the space, not the direction.

Distance from Zero

The absolute value of a number reflects its distance from zero without taking direction into account. On the number line:

• Zero is your reference point.
• Distance is always measured in positive terms, just like counting steps.
• Both \(1.5\) and \(-1.5\) are the same distance away from zero.

So, for the number \(-1.5\), to find its absolute value, envision how many units away from zero it sits. You don’t concern yourself with the direction (whether it’s negative or positive) but purely with the amount of separation from zero. Thus, \( |-1.5| = 1.5 \).Understanding distance from zero helps clarify why the absolute value is always non-negative—it merely counts steps.

Positive Numbers

Positive numbers are those greater than zero and are located to the right of zero on the number line. When dealing with absolute values:

• The result is always a positive number or zero.
• Absolute value measures how far a number is from zero, disregarding negative signs.

Even if your starting point was a negative number like \(-1.5\), once you measure its distance from zero, the result is always positive. That’s why \(|-1.5| = 1.5\).Whether you have a negative or positive starting point, the absolute value translates it into a positive result. This principle is foundational in mathematics, signifying numerical magnitudes independently of directions.