Problem 11 - Precalculus: Functions and Graphs

Question

Rewrite the number without using the absolute value symbol, and simplify the result. (a) \((-5)|3-6|\) (b) \(|-6| /(-2)\) (c) \(|-7|+|4|\)

Step-by-Step Solution

Answer

(a) -15, (b) -3, (c) 11.

Step 1: Identify Absolute Value

First, identify the absolute values in each part of the expression. Absolute value represents the distance of a number from zero on the number line, so it is always positive. For part (a), we have \(|3-6|\). For part (b), we have \(|-6|\). For part (c), we have \(|-7|\) and \(|4|\).

Step 2: Evaluate Absolute Value

Calculate the absolute values for each expression:(a) \(|3-6| = |-3| = 3\) as \-3\ is 3 units from zero.(b) \(|-6| = 6\) because \-6\ is 6 units from zero.(c) \(|-7| = 7\) and \(|4| = 4\) since both numbers are 7 and 4 units from zero, respectively.

Step 3: Rewrite Expression Without Absolute Value

Now replace the absolute values in the expressions with their evaluated positive numbers:(a) \(-5 imes 3\) which gives the new expression.(b) \(6 / -2\) which gives the new expression.(c) \(7 + 4\) which gives the new expression.

Step 4: Simplify the Expression

Simplify each of the rewritten expressions:(a) \(-5 imes 3 = -15\).(b) \(6 / -2 = -3\).(c) \(7 + 4 = 11\).

Key Concepts

Distance from Zero

The concept of absolute value is closely tied to the notion of "distance from zero." Imagine a number line, a simple visual tool where numbers are arranged in order. Absolute value measures how far a number is from zero on this line, always providing a non-negative result. For instance, if we look at the number \( -7 \), we observe that it is 7 spaces to the left of zero. Thus, \(|-7| = 7\). Similarly, for positive numbers, such as 4, the absolute value is simply the number itself, \(|4| = 4\), because it is 4 units from zero.

• For any negative number, the absolute value involves flipping it to its positive counterpart.
• For any positive number, the absolute value remains unchanged.
• Zero's absolute value is zero since it is itself at the starting point.

Absolute value helps us simplify expressions by focusing on distance rather than direction, making complex calculations much easier to handle.

Simplifying Expressions

When simplifying expressions that involve absolute values, the first step is to treat the absolute values as distances from zero. By doing this, we can replace them with their positive equivalents, making calculations straightforward. Let's look at the process:

• Identify any absolute value symbols in the mathematical expression.
• Calculate the distance from zero, replacing each absolute value with its positive result.
• Rewrite the expression without absolute value bars, substituting in your calculated values.

After evaluating, the expression becomes much simpler. Take the expression \[ |-6| /(-2) \]. First, we convert \[ |-6| \] to \ 6 \; based on its distance from zero. Substituting this back, we get \[ 6 / -2\]. The final simplification gives us \-3\, which demonstrates how simplifying can lead to an easy-to-understand result.

Number Line Concepts

A number line is a valuable tool that aids in the understanding of absolute values and distances between numbers. It represents numbers as points spaced out on a line, where zero is the central reference point. Using the number line, we can visually determine the absolute value by counting the steps from zero: to the left for negative numbers, and to the right for positive numbers. Here’s how it applies:

• Addition and subtraction on this line represent moving right and left, respectively.
• The number line helps in visualizing why \( |-3| \) equals 3, as it is 3 steps to the left of zero.
• It clearly shows relations: \(3 + 5 = 8\) as a move further to the right.

Implementing number line concepts makes it clear that absolute values are not merely symbols; they are practical expressions of distance devoid of negative direction, showcasing numbers in their simplest form.