Problem 91 - Algebra A Combined Function

Question

In your own words, explain how to find the absolute value of a number.

Step-by-Step Solution

Answer

The absolute value of a number is its distance from zero; it is always non-negative regardless of whether the number is positive, negative, or zero.

Step 1: Understanding Absolute Value

The absolute value of a number is the distance between that number and zero on the number line. It is always a non-negative number because distance isn't negative.

Step 2: Identifying Number Type

First, identify whether the given number is positive, negative, or zero. The type of number will determine the next steps to finding its absolute value.

Step 3: Finding Absolute Value of Positive Number

If the number is positive, its absolute value is the number itself. For example, the absolute value of 7 is 7.

Step 4: Finding Absolute Value of Negative Number

If the number is negative, multiply the number by -1 to make it positive. For example, the absolute value of -5 is computed as -(-5) = 5.

Step 5: Finding Absolute Value of Zero

If the number is zero, its absolute value is also zero.

Key Concepts

number line

A number line is a visual representation that helps us understand and compare numbers. It is drawn as a horizontal line where numbers are positioned at equal intervals along its length.

Here, zero is located at the center of the number line, with positive numbers increasing to the right and negative numbers extending to the left.

Number lines assist in visually depicting mathematical concepts, like absolute value, by providing a simple way to see the location and relationships between numbers.

• The number line's central point is zero.
• Numbers to the right of zero are positive, while numbers to the left are negative.
• Each point on the number line corresponds to a number.

distance

In mathematics, distance refers to how far apart numbers are from each other, particularly when considered on a number line. It gives a measurable concept to the idea of 'farther' or 'closer'.

When calculating the absolute value of a number, we are essentially looking for the distance of that number from zero on the number line.

Because distance cannot be negative, absolute value is always a non-negative number.

• Distance is always measured in non-negative values.
• It represents how far numbers are from the central point zero.
• Computing distance gives tangible meaning to abstract numbers.

non-negative number

A non-negative number is any number that is either positive or zero. It simply means a number that is not negative.

Absolute value results in a non-negative number because it measures distance.

Imagine taking a step back in math, disregarding the direction, and focusing solely on the size of the step. This is the essence of non-negative values in the context of absolute value.

• Non-negative numbers include zero and positive numbers.
• When working with absolute values, the outcome is always a non-negative integer or real number.
• It helps eliminate the notion of negative magnitude.