Problem 39 - Precalculus

Question

Say whether the statement is TRUE or FALSE. (In Exercises \(37-40\), do not use a calculator or table; use instead the approximations \(\sqrt{2} \approx 1.4 \text { and } \pi \approx 3.1 .)\) $$2 \sqrt{2} \geq 2$$

Step-by-Step Solution

Answer

The statement is TRUE.

Step 1: Understand the Expression

The given expression is \(2 \sqrt{2} \geq 2\). This is an inequality, and we must determine if it is true or false.

Step 2: Use the Approximation

We are given that \(\sqrt{2} \approx 1.4\). Substitute this approximate value into the expression: \(2 \times 1.4\).

Step 3: Calculate the Left Side

Calculate \(2 \times 1.4 = 2.8\). This is the left side of the inequality after substitution.

Step 4: Compare Both Sides of the Inequality

Now we need to compare the calculated left side, 2.8, with the right side of the inequality, which is 2. Since 2.8 is greater than 2, this means the inequality \(2 \sqrt{2} \geq 2\) holds true.

Key Concepts

Inequality Expressions

An inequality expression is a mathematical statement that relates two values, indicating whether one value is greater than, less than, equal to, or not equal to another value. In the given exercise, our task is to determine the truth value of the inequality expression \(2 \sqrt{2} \geq 2\).

Inequality expressions use symbols such as:

• \(>\) greater than
• \(<\) less than
• \(\geq\) greater than or equal to
• \(\leq\) less than or equal to

These expressions can be used to compare numbers, variables, and even complex expressions. Understanding the meanings of these symbols is crucial for correctly interpreting and solving inequality problems.

In this specific problem, our focus was on the "greater than or equal to" sign. This means that the value on the left, \(2 \sqrt{2}\), must be either equal to or greater than the value on the right, 2. By understanding this, we can correctly assess the validity of the inequality.

Approximations

Approximations are often used in mathematics to simplify complex expressions and calculations, making them easier to understand and work with. They are particularly helpful when exact calculations are complex or unnecessary.

In our exercise, the square root of 2 is approximated as \(\sqrt{2} \approx 1.4\). This simplifies the calculation of \(2 \sqrt{2}\) by allowing us to multiply easy numbers instead.
It is important to understand that approximations are not exact but are close representations of the true value. This means they provide us with a useful method to estimate and compare mathematical expressions, especially in inequalities.

When given approximations, always substitute them back into the original expression as shown in the steps of the solution. This can help you to quickly evaluate whether the inequality is true or false without using more precise calculations.

Mathematical Reasoning

Mathematical reasoning involves using logical thinking to solve problems and make sense of mathematical concepts, such as inequalities. When faced with inequality problems, like \(2 \sqrt{2} \geq 2\), it is essential to follow a systematic procedure.

Start by reviewing and understanding the given inequality. Then, substitute any provided approximations to simplify your analysis. In our example, replacing \(\sqrt{2}\) with 1.4 allows us to compute the value more straightforwardly: \(2 \times 1.4 = 2.8\).

The next step is comparing the sides of the inequality: assess whether the left value is indeed greater than or equal to the right value. Here, 2.8 is greater than 2, so the inequality holds true.

Using such methods ensures clarity and precision in reasoning. Always ensure each step logically follows the previous one to reach a valid conclusion. This practice will help you determine the validity of inequalities confidently and accurately.