Problem 71 - Beginning and Intermediate Algebra

Question

Select the lesser of the two given numbers. \(-|-6|,-|-4|\)

Step-by-Step Solution

Answer

-|-6|

Step 1: Understand the Absolute Value Concept

Absolute value, denoted by vertical bars \( | \), represents the distance of a number from zero on a number line without considering direction. Therefore, for any negative number \( -a \), \( |-a| = a \).

Step 2: Calculate the Absolute Values

For the given numbers \( -|-6| \) and \( -|-4| \), first, find their absolute values: \[ |-6| = 6 \] \[ |-4| = 4 \]

Step 3: Apply the Negative Sign

Next, apply the negative sign to the absolute values calculated: \[ -|-6| = -6 \] \[ -|-4| = -4 \]

Step 4: Compare the Negative Values

Finally, compare \( -6 \) and \( -4 \). Since \( -6 \) is less than \( -4 \) (remember that more negative means lesser value), \( -|-6| \) is the lesser number.

Key Concepts

negative numbers

Negative numbers are numbers that are less than zero. They are usually written with a minus sign in front of them, like -1, -2, and so on. Negative numbers represent values that are opposite to positive numbers.
This concept is essential when dealing with real-life situations such as temperatures below zero or debts.
When dealing with negative numbers, it’s important to remember certain properties:

• Adding a negative number is the same as subtracting the corresponding positive number. E.g., 5 + (-3) = 5 - 3.
• Multiplying two negative numbers gives a positive result. E.g., (-2) * (-3) = 6.
• Multiplying a positive number by a negative number gives a negative result. E.g., 4 * (-5) = -20.

number line

A number line is a visual representation of numbers placed in order on a straight line. It helps in understanding the position and order of numbers.
Each point on the line corresponds to a number, with zero usually placed at the center. Numbers to the right of zero are positive while numbers to the left are negative.
Using a number line is very helpful in visualizing concepts like absolute value and comparison of numbers.
For example:

• To represent -4 on a number line, move 4 units to the left of zero.
• To represent +4, move 4 units to the right of zero.

The distance from zero on the number line represents the absolute value, which illustrates the concept without considering the direction.

comparison of integers

Comparing integers means deciding which number is larger or smaller. This can be intuitively done using a number line where the value increases as you move to the right.
For integers, some simple rules help determine their order:

• Any positive number is greater than any negative number. For instance, 3 is greater than -4.
• For negative numbers, the one with a larger absolute value is actually considered lesser. For example, between -4 and -6, -6 is less than -4.
• To compare two positive numbers, simply look at their values. For instance, 5 is greater than 2.

In the given exercise's context, you start by finding the absolute value and then placing them back within their negative context to compare them accurately. As shown:
\[-|-6| = -6 \] \[-|-4| = -4 \] and since -6 is lesser than -4, -|-6| is the lesser number.