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Chapter 1: Problem 80

Explain why the inequality \(|x|>-5\) is true for all real numbers \(x\).

Short Answer

Expert verified
Since \(|x|\) is always non-negative, \( |x| > -5 \) is true for all real numbers \(x\).

Step by step solution

01

- Understand the Concept of Absolute Value

Absolute value \(|x|\) represents the distance of a number \(x\) from zero on the number line. Absolute values are always non-negative, meaning \(|x| \geq 0 \) for any real number \(x\).
02

- Analyze the Given Inequality

The inequality given is \(|x| > -5\). Since the absolute value of any real number is always greater than or equal to zero, it is always going to be greater than \(-5\).
03

- Compare Absolute Value and Negative Number

Noticing that \(|x| > 0\), it logically follows that \(|x|\) will also be greater than any negative number such as \(-5\). This means the inequality is always true regardless of the value of \(x\).

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Key Concepts

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

absolute value
Absolute value is a fundamental concept in mathematics. It refers to the distance of a number from zero on the number line. This distance is always non-negative, meaning it can never be less than zero. The absolute value of a number is denoted by vertical bars around the number, like this: \(|x|\).

For instance:
  • The absolute value of 5 is 5, written as \(|5| = 5\).
  • The absolute value of -7 is 7, written as \(|-7| = 7\).
  • The absolute value of 0 is 0, written as \(|0| = 0\).
These examples show that absolute value ignores whether a number is positive or negative. It only considers how far that number is from zero. Remember, the absolute value \(|x|\) is always zero or a positive number but never negative.
inequalities
Inequalities are statements that show the relationship between two values. Instead of saying two values are equal, inequalities compare their sizes. The main symbols used are:

  • \(< \gt \) - greater than
  • \(< \lt \) - less than
  • \(< \geq \) - greater than or equal to
  • \(< \leq \) - less than or equal to
Consider the inequality \(|x| > -5\). Here, we see the symbol \(|x|\) representing an absolute value. When we say \(|x| > -5\), it means the distance from zero is greater than -5. Because absolute values can never be negative, comparing them to a negative number always holds true. Thus, \(|x|\) will always be greater than -5 for any real number x. This is the power of understanding inequalities in relation to absolute values.
real numbers
Real numbers include all the numbers that we can find on the number line. This comprises:

  • Natural numbers (1, 2, 3, ...)
  • Whole numbers (0, 1, 2, 3, ...)
  • Integers (..., -2, -1, 0, 1, 2, ...)
  • Rational numbers (fractions like 1/2, -3/4)
  • Irrational numbers (numbers like 鈭�2, 蟺)
Real numbers can be positive, negative, or zero. When working with inequalities and absolute values, it is important to realize all real numbers are represented. For any real number x, \(|x| \geq 0\).

In the given problem \(|x| > -5\), any real number x satisfies the inequality, since \(|x|\) is always 0 or more which is certainly more than -5. Thus, these concepts together show that \(|x| > -5\) is universally true in the realm of real numbers.

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Most popular questions from this chapter

Solve the equations. \(\left|\frac{1}{4} w\right|=|4 w|\)

Determine the set of values for \(x\) for which the radical expression would produce a real number. For example, the expression \(\sqrt{x-1}\) is a real number if \(x-1 \geq 0\) or equivalently, \(x \geq 1\). a. \(\sqrt{3 x-7}\) b. \(\sqrt[4]{3 x-7}\)

A rectangular garden is to be constructed so that the width is \(100 \mathrm{ft}\). What are the possible values for the length of the garden if at most \(800 \mathrm{ft}\) of fencing is to be used?

Solve the inequality, and write the solution set in interval notation. \(15<|-2 d-3|+6\)

Suppose that over the course of 3 months, a court system selects 1000 jurors from a large jury pool having an equal number of males and females. If males and females are selected at random, there should be approximately 500 males and 500 females selected. However, these values may vary slightly. If 1000 people are selected at random from the large jury pool, the inequality \(\left|\frac{x-500}{\sqrt{250}}\right|<1.96\) gives the "reasonable" range for the number of women selected, \(x\). a. Solve the inequality and interpret the answer in the context of this problem. (Hint: Round so that the endpoints of the interval are whole numbers, but still within the interval defined by the inequality.) b. If the group of 1000 jurors has 560 women, does it appear that there may be some bias toward women jurors?
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