Problem 73 - Algebra A Combined Function

Question

Insert \(<,>,\) or \(=\) in the appropriate space to make each statement true. $$ |-2| \quad|-2.7| $$

Step-by-Step Solution

Answer

|-2| < |-2.7|

Step 1: Calculate Absolute Values

First, we need to find the absolute value of each number given.- The absolute value of a number is its distance from 0 on the number line, without considering its direction.- The absolute value of \(-2\) is calculated as follows: \(|-2| = 2\)- The absolute value of \(-2.7\) is similarly calculated: \(|-2.7| = 2.7\)

Step 2: Compare Absolute Values

Next, we compare the absolute values obtained:- We found that \(|-2| = 2\) and \(|-2.7| = 2.7\).- Notice that \(2 < 2.7\).

Step 3: Insert the Correct Symbol

Now, insert the correct symbol to make the statement true:- Since \(2 < 2.7\), the relationship between their absolute values is: \(|-2| < |-2.7|\)

Key Concepts

Inequality Comparison

When comparing inequalities, it is essential to understand how we can use symbols like ">", "<", or "=" to compare two numbers or expressions. Inequality symbols help us determine the relative size of the values we are comparing.
For example:

• "<" means "less than"
• ">" means "greater than"
• "=" means "equal to"

In the context of absolute values, inequality comparison allows us to see how different numbers measure up when we ignore their signs.
It helps determine which absolute value is larger or smaller. In our exercise, we compared \(|-2|\) and \(|-2.7|\), finding that \(2 < 2.7\). So we use the "<" symbol to show that \(|-2| < |-2.7|\).
Understanding how to use these symbols correctly helps clarify any mathematical expression or equation.

Absolute Value Properties

Absolute value refers to the distance of a number from zero on the number line, irrespective of direction. Whether the number is positive or negative, its absolute value is always non-negative.
For instance:

• The absolute value of any positive number, say \(5\), is itself: \(|5| = 5\).
• The absolute value of any negative number, say \(-5\), is the positive version of that number: \(|-5| = 5\).

Some critical properties of absolute values include:

• \(\|x\| \geq 0\) for any real number \(x\). Absolute values are never negative.
• The absolute value of zero is zero: \(|0| = 0\).
• For any real numbers \(-a\) and \(a\), \(|-a| = |a|\). The absolute doesn't regard a sign as it measures distance.

Grasping these properties helps you easily solve exercises with absolute values, allowing for accurate comparisons, as seen in comparing \(|-2|\) and \(|-2.7|\), where ignoring the signs, we find their respective distances from zero.

Number Line Representation

A number line offers a visual representation of numbers where each point corresponds to a number. It extends infinitely in both directions, with zero typically placed in the center.
Positive numbers are displayed to the right of zero, and negative numbers to the left.
When discussing absolute values in terms of a number line:

• The distance of the number from zero is the absolute value.
• Points equidistant from zero on either side of the number line have the same absolute value.

Consider the number \(-2\) on the number line: its distance from zero, indicated by \(|-2|\), is two units to the left. Meanwhile, for \(-2.7\), the absolute value shows it is 2.7 units left of zero.
Visualizing these distances can help understand why \(|-2|\) is less than \(|-2.7|\). The smaller the distance on the number line, the smaller the absolute value.
Using a number line to illustrate such exercises simplifies recognizing comparisons of absolute values and inequalities.