Problem 5 - Elementary and Intermediate Algebra

Question

For each pair of numbers, which one has the larger absolute value? a. 6 or 5 b. 8.9 or \(-9.2\)

Step-by-Step Solution

Answer

a. 6; b. -9.2

Step 1: Understand Absolute Value

Absolute value refers to the distance of a number from zero on the number line, regardless of its direction. For any real number \(a\), its absolute value is denoted by \(|a|\). For example, \(|-3| = 3\) and \(|7| = 7\).

Step 2: Calculate Absolute Values for Pair 'a'

For the pair of numbers 6 and 5, we need to find their absolute values: \(|6| = 6\) and \(|5| = 5\).

Step 3: Compare Absolute Values for Pair 'a'

Comparing the absolute values \(6\) and \(5\), we see that the absolute value of 6 is larger than that of 5.

Step 4: Calculate Absolute Values for Pair 'b'

For the pair of numbers 8.9 and \(-9.2\), their absolute values are \(|8.9| = 8.9\) and \(|-9.2| = 9.2\).

Step 5: Compare Absolute Values for Pair 'b'

Comparing the absolute values \(8.9\) and \(9.2\), it is clear that the absolute value of \(-9.2\), which is \(9.2\), is larger than that of 8.9.

Key Concepts

Number Line

The number line is a visual representation of numbers in a straight, horizontal line. It helps us understand the relative positions of numbers by placing them in order, from left to right. The center of the number line is zero, with negative numbers extending to the left and positive numbers to the right.

A few key points to remember about the number line:

• Each point on the number line corresponds to a unique real number.
• The distance between any number and zero represents its absolute value.
• Numbers to the right are always greater than numbers to the left.

Understanding the number line is crucial for visualizing concepts such as addition, subtraction, and absolute value. By locating numbers on this line, we can easily compare their magnitude and see which number is larger or smaller.

Real Numbers

Real numbers include all the numbers that you can find on the number line. This means both numbers that we use every day in calculations, like whole numbers and decimals, and more complex numbers like irrational numbers.

Here are some categories of real numbers:

• Natural numbers: 1, 2, 3, and so on.
• Whole numbers: 0, 1, 2, 3, and so on.
• Integers: ..., -3, -2, -1, 0, 1, 2, 3, ...
• Rational numbers: Numbers that can be expressed as the quotient or fraction of two integers, such as 1/2 or -3/4.
• Irrational numbers: Numbers that cannot be expressed as a simple fraction, like \(\pi\) or the square root of 2.

Real numbers are essential for every type of measurement and calculation. Understanding real numbers helps us grasp the vastness of different quantities and magnitudes.

Compare Numbers

Comparing numbers involves determining which of two numbers is larger or smaller. This skill is important for solving many types of math problems, including those involving absolute values.

To compare numbers:

• Think about their positions on the number line. Numbers further to the right are larger.
• Consider their absolute values; sometimes, larger positive or negative numbers have larger absolute values.
• Use inequalities to express comparisons: < or > indicate if one number is smaller or larger than another.

For instance, comparing numbers 6 and 5 involves checking if 6 is greater than 5, which it is, as 6 is to the right. Similarly, \(\-9.2\) on the number line is further from zero than 8.9, hence its absolute value is larger. Understanding the principles of number comparison is key in many mathematical analyses and can simplify the problem-solving process significantly.