Problem 34 - 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 \(\sqrt{2} \approx 1.4 \text { and } \pi \approx 3.1 .)\) $$\sqrt{7}-2 \geq 0$$

Step-by-Step Solution

Answer

TRUE

Step 1: Use Approximation for \(\sqrt{7}\)

To solve this problem, we need an approximation for \(\sqrt{7}\). We know that \(\sqrt{4} = 2\) and \(\sqrt{9} = 3\), hence \(\sqrt{7}\) must be between 2 and 3. A closer approximation is that \(\sqrt{7} \approx 2.6\).

Step 2: Calculate \(\sqrt{7} - 2\)

Using the approximation from Step 1, substitute it in the expression: \(\sqrt{7} - 2 \approx 2.6 - 2\). This simplifies to 0.6.

Step 3: Compare \(\sqrt{7} - 2\) with Zero

The expression \(2.6 - 2 = 0.6\) is positive, meaning \(\sqrt{7} - 2 \geq 0\) is true.

Key Concepts

Square Root Approximations

Square root approximations allow us to estimate the value of a square root without using a calculator. In this exercise, we need to approximate \( \sqrt{7} \). We know that \( \sqrt{4} = 2 \) and \( \sqrt{9} = 3 \). This means \( \sqrt{7} \) is somewhere between these two values. By observing commonly known approximations, we can further narrow this down to \( \sqrt{7} \approx 2.6 \).
Approximations are useful because they make difficult calculations manageable. With a bit of estimation, you can compare or perform operations with numbers without needing precise calculations. This skill is particularly useful in exams where calculators are not allowed.
Approximating square roots is about looking for known values around your desired square root. For instance, knowing that \( \sqrt{9} = 3 \), and \( \sqrt{7} \) is slightly less than 3, gives us a good starting point.

Mathematical Reasoning

Mathematical reasoning is the process of using logical thinking to solve problems. It is critical when evaluating statements like \( \sqrt{7} - 2 \geq 0 \). The process involves several steps:

• Recognizing what is known or given (like \( \sqrt{4} = 2 \) and \( \sqrt{9} = 3 \)).
• Making a logical estimation based on that knowledge, like assuming \( \sqrt{7} \approx 2.6 \).
• Substituting the approximation into your expression to evaluate it, such as \( 2.6 - 2 \).
• Drawing a conclusion from this simplified value to determine whether the original statement is true.

In our example, the logical deductions lead us step-by-step from estimated values to a final conclusion about the inequality. This systematic approach ensures you understand each component of the problem and validate each step.

Inequality Comparisons

Inequality comparisons are about understanding the relationship between two expressions or values. Here, we looked at whether \( \sqrt{7} - 2 \geq 0 \), which involves comparing \( \sqrt{7} \) and \( 2 \).
We used the approximation of \( \sqrt{7} \approx 2.6 \). The expression thus becomes \( 2.6 - 2 \), simplifying this to \( 0.6 \).
This simplification shows that \( 0.6 \) is indeed greater than or equal to 0, meaning the inequality is true.

• Inequality symbols like \( \geq \) indicate that a value on the left is either greater than or the same as the one on the right.
• Inequality comparisons help in determining the nature of mathematical relationships between variables.

Grasping how to perform and interpret these comparisons is essential for solving algebra problems accurately.