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Chapter 3: Problem 49

For all real numbers \(x,|-x|=|x|\).

Short Answer

Expert verified
We proved that for all real numbers x, \(|-x|=|x|\) by considering two cases: 1. If x is positive, then we showed that \(|-x| = -(-x) = x = |x|\). 2. If x is negative or zero, then we showed that \(|-x| = -x = -(-x) = |x|\). Hence, in both cases, \(|-x|=|x|\) holds true for all real numbers x.

Step by step solution

01

Case 1: When x is positive

If x is a positive real number, then -x will be negative. In this case, according to the definition of the absolute value, we have: \(|-x| = -(-x) = x\) Since x is positive, the definition of the absolute value gives us: \(|x| = x\) So, in this case, we have \(|-x| = |x|\).
02

Case 2: When x is negative or zero

If x is a negative real number or zero, then -x will be positive (or zero). In this case, according to the definition of the absolute value, we have: \(|-x| = -x\) Since x is negative (or zero), the definition of the absolute value gives us: \(|x| = -(x)\) So, in this case, we have \(|-x| = |x|\).
03

Conclusion

In both cases, we have shown that \(|-x|=|x|\). Therefore, for all real numbers x, \(|-x|=|x|\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Real Numbers
Real numbers encompass both rational and irrational numbers. They include all the numbers on the number line, like whole numbers, fractions, and decimals. Real numbers can be either positive, negative, or zero. Understanding real numbers is crucial because they form the basis for various mathematical principles and everyday calculations.
Real numbers include a wide range of values:
  • Integers: These are whole numbers that can be positive, negative, or zero, such as -3, 0, or 7.
  • Rational numbers: These numbers can be expressed as fractions, like 1/2 or 0.75.
  • Irrational numbers: Numbers that cannot be written as simple fractions, such as 蟺 or 鈭�2.
In this exercise, understanding that any real number can be positive, negative, or zero helps us analyze the behavior of expressions like absolute values.
Mathematical Proof
Mathematical proof is a logical method used to demonstrate that a particular statement or theorem is universally true. In mathematics, a proof ensures that a statement follows logically from a set of axioms or previously established truths.
In the exercise example, we want to prove that for any real number \(x\), the absolute value of \(-x\) equals the absolute value of \(x\). Here, the proof involves showing that, regardless of whether \(x\) is positive, negative, or zero, the statement holds true:
  • Logical consistency: Each case needs to be considered to guarantee the accuracy of the proof.
  • Clear reasoning: Making connections between known principles, like the definition of absolute value, helps to construct the proof by demonstrating relationships and implications.
A comprehensive proof covers all possible scenarios, ensuring that nothing is left to guesswork.
Case Analysis
Case analysis is a technique used in mathematics to handle different possible situations in a problem, by breaking it down into cases and analyzing each one individually.
This approach was used in the exercise to prove that \(|-x| = |x|\) for all real numbers \(x\). By dividing the problem into two cases:
  • Case 1: \(x\) is positive.
  • Case 2: \(x\) is negative or zero.
Each case was examined to show that the equality holds, regardless of whether \(x\) is positive, negative, or zero. This ensures a complete proof by systematically checking all potential scenarios.
Algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It is about finding the unknown from known quantities by using letters and symbols to represent numbers and quantities in formulas and equations.
In this problem, algebraic manipulation helps us explore the absolute value expression:\[|-x| = |x|\]The steps involve using definitions and properties of absolute value to solve for different cases:
  • Definition Usage: Applying the definition of absolute value correctly is key. For example, \(|-x|\) results in \(-x\) or \(x\) depending on the sign of \(x\).
  • Simplification: In case 1, when \(x\) is positive, \(|x| = x\), and when \(x\) is negative, \(|x| = -x\).
Through these algebraic manipulations, the solution shows how to prove that the absolute value holds for all real numbers \(x\). Understanding basic algebra principles is essential to grasp such transformations and simplifications.

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Most popular questions from this chapter

If \(n\) is an integer and \(n>1\), then \(n !\) is the product of \(n\) and every other positive integer that is less than \(n\). For example, \(5 !=5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\). a. Write \(6 !\) in standard factored form. b. Write \(20 !\) in standard factored form. c. Without computing the value of \((20 !)^{2}\) determine how many zeros are at the end of this number when it is written in decimal form. Justify your answer.

For all integers \(m, m^{2}=5 k\), or \(m^{2}=5 k+1\), or \(m^{2}=\) \(5 k+4\) for some integer \(k\).

"Proof: Suppose \(r\) and \(s\) are rational numbers. By definition of rational, \(r=a / b\) for some integers \(a\) and \(b\) with \(b \neq 0\), and \(s=a / b\) for some integers \(a\) and \(b\) with \(b \neq 0\). Then \(r+s=a / b+a / b=2 a / b\). Let \(p=2 a\). Then \(p\) is an integer since it is a product of integers. Hence \(r+s=p / b\), where \(p\) and \(b\) are integers and \(b \neq 0\). Thus \(r+s\) is a rational number by definition of rational. This is what was to be shown."

When an integer \(a\) is divided by 7 , the remainder is 4 . What is the remainder when \(5 a\) is divided by \(7 ?\)

Definition: The least common multiple of two nonzero integers \(a\) and \(b\), denoted \(\operatorname{lcm}(a, b)\), is the positive integer \(c\) such that a. \(a \mid c\) and \(b \mid c\) b. for all integers \(m\), if \(a \mid m\) and \(b \mid m\), then \(c \mid m\). Prove that for all positive integers \(a\) and \(b, a \mid b\) if, and only if, \(\operatorname{lcm}(a, b)=b\).
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