Problem 81 - Contemporary Precalculus

Question

Fill the blank with \(<,=\), or \(>\) so that the resulting statement is true. |3| __________-|4|

Step-by-Step Solution

Answer

Question: Fill in the blank with the appropriate symbol (<, =, or >): \(|3|\) ___ \(|-4|\) Answer: \(|3| < |-4|\)

Step 1: Write down the given numbers with absolute value bars

We are given two numbers: \(|3|\) and \(|-4|\).

Step 2: Determine the absolute values

Find the absolute values of the given numbers: \(|3| = 3\) (since 3 is already a positive number) \(|-4| = 4\) (the absolute value of a negative number is its positive counterpart)

Step 3: Compare the absolute values

Compare the absolute values we found in Step 2: \(3\) and \(4\) Since \(3 < 4\), we can now complete the statement: \(|3| < |-4|\)

Key Concepts

Inequalities

Inequalities are a way to express that two values are not equal, and instead one is greater or less than the other. In mathematics, inequalities are often represented using symbols like \(<\), \(=\), or \(>\). These symbols help us compare two numbers or expressions. When we see \(<\), it means the value on the left is smaller;
\(>\) means the value on the left is bigger; and \(=\) indicates both sides are the same.

• \(<\) means 'less than'
• \(>\) means 'greater than'
• \(=\) means 'equal to'

Applying this to our example with absolute values, we compared \(|3|\) and \(|-4|\). By using absolute values, we turned these into 3 and 4.
Since 3 is less than 4, the inequality symbol we used is \(<\). Therefore, \(|3| < |-4|\).
Understanding inequalities allows us to determine and express relationships between numbers in both simple and complex mathematical scenarios.

Real Numbers

Real numbers include all the numbers you can think of along a number line. They encompass natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Real numbers can be positive, zero, or negative.

• Natural numbers: 1, 2, 3, ...
• Whole numbers: 0, 1, 2, 3, ...
• Integers: ..., -3, -2, -1, 0, 1, 2, 3, ...
• Rational numbers: fractions or ratios, like 1/2, 2/3, 5/1
• Irrational numbers: numbers that can't be expressed as a simple fraction, like \(\sqrt{2}\) or \(\pi\)

The concept of real numbers is crucial because it helps us define and understand the scope of numbers we deal with in real-life situations. In our problem, both 3 and -4 are real numbers. They fall within this broad category that includes all possible continuous numerical values you might encounter in reality.
This universality allows us to perform arithmetic operations and comparisons between them, such as the inequality comparison in our exercise.

Comparison of Magnitudes

Comparison of magnitudes involves determining which number is larger or smaller after removing any negative signs. Absolute value is a tool used for this purpose. By converting numbers to their absolute values, we focus purely on their size without considering direction on a number line.

• Absolute value: Converts any number to its non-negative form
• Magnitude: The size or amount of a number, irrespective of its sign

Let's break it down with the example: we have two numbers, 3 and -4. Their absolute values are 3 and 4 respectively.
So, we compare these values: 3 and 4 are the magnitudes. Seeing that 3 is smaller, we conclude \(3 < 4\). This is how we use absolute values to determine relative size without the effect of negative signs.
Understanding magnitude comparison is important in math for placing numbers in order, selecting the smallest or largest numbers from a set, and many other tasks in fields like statistics, finance, and even science.