Problem 57 - College Algebra and Calculus: An Applied Approach

Question

Place the correct symbol \((<,>\), or \(=)\) between the two real numbers.\(|-3| \quad-|-3|\)

Step-by-Step Solution

Answer

The correct symbol to place between \(|-3|\) and \(-|-3|\) is '>'. So, \(|-3| > -|-3|\).

Step 1: Evaluate First Absolute Value

First, resolve the absolute value of -3, which is \(|-3|\). The absolute value of a negative number is the number without its negative sign, so \(|-3|\) equals 3.

Step 2: Evaluate Second Absolute Value

Next, resolve the absolute value of -3, which is \(-|-3|\). The absolute value of -3 is 3, and considering the negative sign outside the absolute symbol, \(-|-3|\) equals -3.

Step 3: Compare Values

Finally, compare the two resolved values, 3 and -3. Since 3 is greater than -3, the correct symbol to place between the two numbers would be '>'.

Key Concepts

Absolute Value

Absolute value is a fundamental concept in mathematics, depicted by vertical bars enclosing a number, such as \(|x|\). It represents the distance of that number from zero on the number line. For instance, the absolute value of both 3 and -3 is 3. This is because distance is always positive, and it measures how far away 3 or -3 is from 0.

Here are some important points about absolute value:

• The absolute value of any positive number is the number itself.
• The absolute value of any negative number is its positive counterpart.
• The absolute value of 0 is 0.

Understanding absolute value can help simplify the process of comparing numbers and solving equations involving distances. In our original exercise, the key task was recognizing that \(|-3|\) equals 3, removing the need to focus on whether the original number was negative or not.

Real Numbers

Real numbers encompass the set of all the numbers that can be found on the number line. This includes a wide range from rational numbers, like fractions and integers, to irrational numbers which cannot be expressed as fractions.

Rational numbers are numbers that can be written as \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b eq 0\). Examples include integers like -3, 0, and 5, and fractions like \(\frac{1}{2}\).

Irrational numbers are numbers that cannot be neatly written as fractions of two integers. Examples include numbers like \(\pi\) and \(\sqrt{2}\), which have non-repeating, non-terminating decimal expansions.

• Real numbers are uncountably infinite.
• Every number that has a position on the number line is a real number.
• They can be used in various mathematical operations such as addition, subtraction, and comparison.
• Absolute values, like \(|-3|\), are always real numbers.

In the provided exercise, both values \(|-3|\) and \(-|-3|\) are real numbers, and understanding their placement on the number line aids in comparing them effectively.

Inequalities

Inequalities express the relationship between two expressions that are not equal. In mathematics, inequalities are used to compare numbers, often using symbols like \(<\), \(>\), and \(\leq\) or \(\geq\) to convey a less than, greater than, or less than or equal relationship.

When comparing two numbers:

• Use \(a < b\) when \(a\) is less than \(b\).
• Use \(a > b\) when \(a\) is greater than \(b\).
• Use \(a = b\) when \(a\) is equal to \(b\).

Understanding inequalities is crucial for solving equations and performing operations involving range or limits. In our original exercise, after calculating the absolute values, the inequality \(3 > -3\) was established, demonstrating how 3 is greater than -3, which helped in placing the correct symbol \(>\) between the compared numbers.