FGV SP Real Numbers Inequality Analysis Find The True Statement

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Hey guys! Let's dive into a fascinating problem from FGV SP that deals with real numbers and inequalities. This is a classic math question that tests our understanding of fundamental properties. So, buckle up, and let's break it down step by step!

The Question

We are given real numbers a, b, and c, and our mission is to identify the true statement among the options provided. The options involve comparisons and operations on these real numbers, focusing on inequalities.

[Original Question]

Sejam a, b e c números reais quaisquer, assinale a afirmação verdadeira:

a) a > b ⇔ a^{2} > b^{2}

b) a > b ⇔ ac > bc

c) √(a^2 + b^2) ≥ a

Let's analyze each option carefully to pinpoint the one that holds true for all real numbers a, b, and c.

Analyzing the Options

Option A: a > b ⇔ a² > b²

This statement claims that a is greater than b if and only if a squared is greater than b squared. To assess this, let’s consider a counterexample. Imagine a = 1 and b = -2. Here, a is greater than b (1 > -2). However, when we square them, we get a² = 1 and b² = 4. In this case, a² is not greater than b² (1 < 4). Therefore, option A is not universally true, as it fails for negative numbers.

The statement fails when dealing with negative numbers. The squaring operation can change the direction of the inequality depending on the signs of a and b. To make this statement true, we would need to add the condition that both a and b are non-negative. It's crucial to consider such edge cases when dealing with inequalities. When squaring both sides of an inequality, the signs of the numbers play a vital role, and overlooking this can lead to incorrect conclusions. Remember, the devil is in the details, and a single counterexample is enough to disprove a statement in mathematics. Always look for cases where the statement might not hold true.

Option B: a > b ⇔ ac > bc

This option suggests that a being greater than b is equivalent to ac being greater than bc. However, this isn't always true. The crucial factor here is the sign of c. If c is positive, the inequality holds. But, if c is negative, multiplying both sides of the inequality by c reverses the inequality sign. For instance, if a = 2, b = 1, and c = -1, then a > b (2 > 1), but ac = -2 and bc = -1, so ac < bc (-2 < -1). This illustrates that the statement is false when c is negative.

The impact of multiplying an inequality by a negative number is a fundamental concept. Many students stumble upon this point, so it’s essential to understand it thoroughly. In our example, the negative value of c flips the direction of the inequality, demonstrating why this option is not universally true. A more accurate statement would include a condition based on the sign of c. For instance, if c > 0, then a > b implies ac > bc. Conversely, if c < 0, then a > b implies ac < bc. This nuanced understanding is key to solving inequality problems correctly. Always remember to consider the sign of the multiplier when working with inequalities. Ignoring this can lead to significant errors in your solutions.

Option C: √(a² + b²) ≥ a

Now, let’s examine option C: √(a² + b²) ≥ a. This statement looks promising! We know that squaring any real number results in a non-negative value. Thus, a² and b² are both non-negative. Consequently, a² + b² will always be greater than or equal to a². Taking the square root of both sides, we get √(a² + b²) ≥ √(a²). Since the square root of a² is the absolute value of a (|a|), we have √(a² + b²) ≥ |a|. The absolute value of any number is always greater than or equal to the number itself, meaning |a| ≥ a. Therefore, √(a² + b²) ≥ a is indeed true for all real numbers a and b.

This inequality holds due to the properties of square roots and squares of real numbers. The key here is to recognize that a squared value is always non-negative. Adding a non-negative number (b²) to a² will only increase the value or keep it the same. The square root function preserves this inequality. Furthermore, understanding the relationship between the square root of a square and the absolute value is crucial. The absolute value ensures that the result is always non-negative, which is why |a| ≥ a always holds. This option highlights the importance of understanding the interplay between different mathematical concepts, such as squares, square roots, and absolute values. Such insights allow us to dissect complex inequalities and arrive at the correct conclusion. This thorough analysis confirms that Option C is the correct answer.

The Verdict

After a thorough analysis, we've determined that option C is the only statement that holds true for all real numbers a, b, and c. Options A and B have counterexamples that prove their falsity.

Key Takeaways

  • Counterexamples are powerful: They can quickly disprove a statement.
  • Sign matters: Be mindful of the signs of numbers, especially when multiplying inequalities or squaring numbers.
  • Fundamental properties are your friends: Leverage the properties of real numbers, squares, square roots, and absolute values.

Conclusion

So there you have it, guys! We've successfully navigated this FGV SP problem by carefully analyzing each option and using our understanding of real numbers and inequalities. Remember to always think critically and look for potential pitfalls when dealing with mathematical statements. Keep practicing, and you'll become inequality masters in no time! This kind of problem reinforces the importance of a solid foundation in mathematical principles. By understanding the fundamentals, you can tackle even the most challenging questions with confidence. Keep up the great work!