Problem 72 - Beginning and Intermediate Algebra

Question

Select the lesser of the two given numbers. \(-|-2|,-|-3|\)

Step-by-Step Solution

Answer

-|-3| is lesser.

Step 1 - Understand Absolute Values

Firstly, recall what the absolute value function does. The absolute value of a number is its distance from zero on the number line, regardless of direction, denoted as \(|x|\). For example, \(|-2| = 2\) and \(|-3| = 3\).

Step 2 - Calculate Inner Absolute Values

Calculate the absolute values within the given expressions: \(|-2| = 2\) and \(|-3| = 3\).

Step 3 - Apply Negations

Apply the negation to the absolute values you calculated: \(-|2| = -2\) and \(-|3| = -3\).

Step 4 - Compare the Results

Now compare the two values: \(-2\) and \(-3\) to determine which one is lesser. Since \(-3\) is less than \(-2\), \(-3\) is the smaller number.

Key Concepts

Absolute Value Properties

The absolute value of a number is how far that number is from zero on the number line, without considering direction. The symbol for absolute value is two vertical bars around a number, such as \(|x|\). Here are some key properties of absolute value:

\[ |a| \geq 0 \]This means that the absolute value of any number is always non-negative.

\[ |a| = |-a| \]This property tells us that the absolute value of a number is the same as its opposite. For example, \(|-5| = 5\) and \(|5| = 5\).

\[ |a \cdot b| = |a| \cdot |b| \]The absolute value of a product is the product of the absolute values. For instance, \(|2 \cdot -4| = |2| \cdot |-4| = 8\).

Understanding these properties helps us simplify expressions involving absolute values, such as in our exercise where \(|-2| == 2\) and \(|-3| == 3\).

Number Line

A number line is a visual representation of numbers stretched out on a straight line.
Zero is located at the center, with positive numbers to the right and negative numbers to the left.
Absolute value represents the distance of a number from zero, without focusing on direction.
For example, on a number line:

• The distance between -2 and 0 is 2 units.
• The distance between -3 and 0 is 3 units.

Visualizing this makes it easier to understand why \(|-2| == 2\) and \(|-3| == 3\).
By plotting numbers on a number line, it simplifies the comparison of values and their absolute equivalents.

Negation of Absolute Value

Negation involves changing the sign of a number.
When negating an absolute value, \-|x|\, you are essentially taking the positive distance and converting it to a negative value.
Let's look at our exercise:

• Calculate \(|-2| == 2\)
• Then negate: \(-|2| == -2\)

Similarly, for the second part:

• Calculate \(|-3| == 3\)
• Then negate: \(-|3| == -3\)

After finding \-2\ and \-3\, we compare them to see which is lesser.
In this context, \-3\ is smaller because it is further to the left on the number line.