Problem 33 - 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 \(\sqrt{2} \approx 1.4 \text { and } \pi \approx 3.1 .)\) $$-2 \leq-2$$

Step-by-Step Solution

Answer

TRUE

Step 1: Interpret the Inequality

The inequality presented is \[-2 \leq -2\].This reads as '-2 is less than or equal to -2'. To determine if this statement is TRUE or FALSE, we need to verify if -2 is indeed less than or equal to -2.

Step 2: Analyze the Equality Part

Since the inequality involves 'less than or equal to', we must check both parts: the 'less than' and the 'equal to'. If the number on the left is equal to the number on the right, then the 'equal to' part of the inequality satisfies the condition. In this case, -2 is equal to -2.

Step 3: Conclude the Analysis

Since -2 equals -2, the condition '-2 is less than or equal to -2' holds true. Therefore, the given inequality is satisfied.

Key Concepts

Understanding Inequality Interpretation

The concept of interpreting inequalities requires us to understand what is being compared. In the exercise, the inequality given is \(-2 \leq -2\). This means we are asked to determine if \(-2\) is either less than or equal to \(-2\). The symbol \(\leq\) combines two ideas:

• Less than: one number is smaller than another.
• Equal to: both numbers are the same.

Therefore, interpreting this inequality involves checking whether \(-2\) matches either of these conditions with itself. Understanding this duality is essential because an inequality with the "or equal to" component requires both parts to be evaluated.

Ways to Ensure Inequality Verification

Inequality verification means confirming if the given statement is true. For the inequality \(-2 \leq -2\), we need to check whether \(-2\) is truly less than or equal to \(-2\).Since the numbers are identical, the concept of "less than" does not apply here. However, they are certainly equal.

• This satisfies the "equal to" condition in the \(\leq\) symbol.

Thus, during verification, if at least one part (either "less than" or "equal to") of the inequality holds true, then the entire inequality is confirmed as true. In this specific situation, the verification shows that the statement is indeed correct, as the condition of being equal is satisfied.

Steps to Find Inequality Solution

Solving an inequality involves several straightforward steps:1. **Read and Understand the Problem:** Begin by reading the inequality carefully. For example, in our case \(-2 \leq -2\).2. **Identify the Components:** Recognize the "less than or equal to" symbol \(\leq\), understanding you are dealing with both inequality and equality.3. **Conduct a Comparative Analysis:** Check each part: - Can the first number truly be less than the second? In this exercise, this isn’t possible since \(-2\) is not less than \(-2\). - Are they equal? Yes, \(-2\) is equal to \(-2\).4. **Conclude the Verification:** Based on your analysis, if either condition ("less than" or "equal to") holds true, then the entire inequality is true.Following these steps ensures you analyze the inequality thoroughly and arrive at the correct conclusion. In this instance, acknowledging the equality between \(-2\) and \(-2\) leads to confirming the inequality as accurate.