Problem 91 - College Algebra

Question

insert either \(<\) or \(>\) in the box between the numbers to make the statement true. $$ \sqrt{2} \square 1.5 $$

Step-by-Step Solution

Answer

The correct inequality is \(\sqrt{2} < 1.5\).

Step 1: Calculate the square root of 2

First, calculate the square root of 2 using a calculator. The square root of 2 is approximately 1.414.

Step 2: Compare the Values

With the calculated value for \(\sqrt{2}\) which is 1.414 and the given 1.5, it is clear that 1.414 is less than 1.5.

Step 3: Insert the Correct Inequality Symbol

Since 1.414 (which is \(\sqrt{2}\)) is less than 1.5, the correct inequality symbol to insert into the box is \(<\). Therefore, \(\sqrt{2} < 1.5\) is the correct inequality.

Key Concepts

Square Roots

Understanding the concept of square roots is essential in mathematics, especially when comparing numbers and solving inequalities. A square root of a number 'x' is a value that, when multiplied by itself, gives 'x'. For instance, in the exercise \(\sqrt{2}\), we're looking for a number that, when squared, equals 2.

However, square roots can often be irrational numbers, meaning they can't be expressed as a simple fraction and their decimal representation goes on forever without repeating. Since the square root of 2 is an irrational number, we use an approximate value for comparisons, such as 1.414. Remember, the level of precision can affect the outcome of comparisons, so it's essential to ensure that the approximation is sufficiently accurate for the context it’s being used in.

Comparing Numbers

Comparing numbers is a fundamental skill that allows us to understand numerical relationships. Whether dealing with whole numbers, fractions, decimals, or even irrational numbers, the ability to judge which is greater or lesser is central to many mathematical concepts such as ordering, inequalities, and functions. When we compare numbers, such as in the exercise \(\sqrt{2} \square 1.5\), it's vital to understand the place value and numeric value.

Visual Representation

Visual aids can also assist in comparing numbers, using a number line can be helpful. For approximate comparisons, rounding the numbers to the nearest whole number or significant figure may simplify the process. It's crucial to remain consistent in the level of precision when comparing numbers to maintain accuracy.

Mathematical Inequalities

Mathematical inequalities are statements that relate two values showing that one value is less than, greater than, less than or equal to, or greater than or equal to another value. The symbols used for these relationships are \( < \), \( > \), \( \leq \) and \( \geq \), respectively. Inequalities are fundamental in expressing the range of possible solutions for an equation or the relationship between different quantities.

Assignment of Correct Inequality

Setting up the correct inequality symbol, as we did in the exercise, requires careful consideration of the relationship between the two numbers. In some cases, additional steps to isolate the variable may be needed when the inequality involves algebraic expressions. Once the numerical comparison is clear, the appropriate symbol is then used to accurately represent the relationship.