Problem 58 - Introductory and Intermediate Algebra for College Students 4th

Question

Insert either \(<\) or \(>\) in the shaded area between each pair of numbers to make a true statement. $$\sqrt{3}\quad\square\quad2$$

Step-by-Step Solution

Answer

\(\sqrt{3} < 2\)

Step 1: Identify the numbers

The numbers given in the exercise are \(\sqrt{3}\) and 2.

Step 2: Determine the approximate or exact value of the square root

The square root of 3, \(\sqrt{3}\), is approximately 1.7320508. This number is less than 2.

Step 3: Compare the numbers

Since the value of \(\sqrt{3}\) is less than 2, the inequality symbol that makes the statement true is \(<\). Therefore, \(\sqrt{3} < 2\).

Key Concepts

Inequalities

In mathematics, inequalities are used to compare two values or expressions. They indicate the relative size or order of two numbers. Inequalities can be represented using symbols such as \(<\) for "less than," and \(>\) for "greater than." When you see an inequality, you are looking at a statement where one value is not equal to the other, but instead, it is either smaller or larger. For example, when we say \(a < b\), it tells us that \(a\) is smaller than \(b\).

There are different types of inequalities:

• Strict inequalities: These include \(<\) and \(>\), indicating one value is strictly less or greater than the other.
• Non-strict inequalities: These include \(\leq\) (less than or equal to) and \(\geq\) (greater than or equal to).

Understanding inequalities is crucial, as they form the foundation for solving a wide range of mathematical problems. They help us analyze and interpret mathematical relationships in both pure and applied mathematics.

Square roots

A square root is a mathematical function that finds a number which, when multiplied by itself, gives the original number. The square root of a number \(x\) is denoted as \(\sqrt{x}\). For example, the square root of 9 is 3 because 3 multiplied by itself equals 9.

When dealing with square roots, you often end up with irrational numbers, especially when the number isn't a perfect square. An irrational number cannot be accurately expressed as a simple fraction, and its decimal form goes on forever without repeating.

In our example, the square root of 3, \(\sqrt{3}\), is approximately 1.7320508. Each irrational square root has a unique decimal expansion. Familiarizing yourself with approximate values of common square roots can make it easier to quickly estimate and compare numbers in various mathematical problems.

Comparison of numbers

Comparing numbers is a fundamental skill in mathematics, essential for understanding how numbers relate to each other. This concept is especially important when dealing with inequalities, as it allows us to determine which number is greater or lesser relative to another.

When comparing two numbers, especially when one or both are irrational (like \(\sqrt{3}\)), it helps to know their approximate decimal values. This allows for an easier and more intuitive comparison. For instance, knowing \(\sqrt{3} \approx 1.73\) simplifies the process of comparing it to a whole number, such as 2.

The key steps in comparison are:

• Evaluate or estimate the approximate values of the numbers involved.
• Compare these values to determine the relationship (less than, greater than, or equal to).

This understanding doesn't just help in exercises like inserting an inequality symbol. It also underpins broader mathematical concepts and is vital in many real-world applications, such as in calculating distances, probabilities, and optimizing solutions.