Problem 77 - Calculus with Concepts in Calculus

Question

If \(x^{2} \leq 25\), is it necessarily true that \(x \leq 5 ?\) Explain.

Step-by-Step Solution

Answer

No, because x can be any value between -5 and 5.

Step 1: Understanding the Given Inequality

We start with the inequality \(x^2 \leq 25\). This means that \(x^2\) can be any value from 0 up to 25, inclusive. Our goal is to determine whether this implies \(x \leq 5\).

Step 2: Solving the Inequality for x

To find the range of values for \(x\), we take the square root of both sides of the inequality \(x^2 \leq 25\). The square root gives two potential ranges: \(-\sqrt{25} \leq x \leq \sqrt{25}\), which simplifies to \(-5 \leq x \leq 5\).

Step 3: Analysis of the Solutions

The range \(-5 \leq x \leq 5\) suggests that \(x\) can be any value between -5 and 5, inclusive. Thus, while it is true that \(x\) can be less than or equal to 5, \(x\) can also be greater than -5, meaning \(x\) does not need to be strictly less than or equal to 5.

Step 4: Conclusion

The given inequality \(x^2 \leq 25\) implies that \(x\) is between \(-5\) and \(5\). Therefore, \(x \leq 5\) is not necessarily true because \(x\) could be negative and still satisfy the inequality.

Key Concepts

Square Root Method

When dealing with inequalities involving squares, the square root method is your go-to tool. It helps simplify and find solutions for cases like those involving the inequality \( x^2 \leq 25 \). By taking the square root of both sides, you unveil the possible numbers for \( x \) that satisfy the inequality. In this example, take the square root of 25, revealing \( \sqrt{25} = 5 \). Remember, when dealing with squares and roots, you must consider both positive and negative solutions.

This means:

• \(-\sqrt{25} \leq x \leq \sqrt{25} \)
• Simplifying gives: \(-5 \leq x \leq 5 \)

By finding these bounds, we understand that \( x \) can take on any value between -5 and 5. This wide range covers many possibilities for \( x \), not just positive values. The square root method broadens the scope of solutions, emphasizing the need to consider all potential solutions, both positive and negative.

Range of Values

Understanding the range of values for \( x \) in inequalities like \( x^2 \leq 25 \) is crucial. The result \(-5 \leq x \leq 5 \) indicates that \( x \) can represent a whole spectrum of numbers between -5 and 5. This conception goes beyond merely observing any specific value and includes any potential number within the boundary.

In this scenario:

• The interval starts from -5 and increases up to 5
• All numbers, both integers and decimals between these bounds, are viable solutions

When considering this range, it's important to remember that each number in this set could represent a solution to the initial inequality. By understanding how to determine the range of values, you gain better insight into the complete solution context, ensuring you consider all possible answers.

Analysis of Solutions

When analyzing solutions to inequalities, such as \( x^2 \leq 25 \), deeper understanding emerges. In this examination, the derived range \(-5 \leq x \leq 5\) underscores a deeper point: \( x \leq 5 \) is only a partial view of the potential values. In reality, \( x \) could come from anywhere within the defined range.

Here’s why this is significant:

• While \( x \leq 5 \) is valid, it’s not exclusive
• Values such as \( x = -3 \) also fit within the range

The comprehensive analysis reveals that simply sticking to positive values misses the completeness of possible solutions. Considering all allowable values assures a full grasp of how inequalities manifest solutions. By exploring both upward and downward trends within the bounds, you construct a more expansive understanding of potential outcomes.