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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

Peggy competes in a biathlon by running and bicycling around a large loop through a city. She runs the loop one time and bicycles the loop five times. She can run \(8 \mathrm{mph}\) and she can ride \(16 \mathrm{mph}\). If the total time it takes her to complete the race is 1 hr \(45 \mathrm{~min}\), determine the distance of the loop.

A ___________ equation is a first-degree equation of the form \(a x+b=0\) where \(a \neq 0\).

Determine if the equation is linear, quadratic, or neither. If the equation is linear or quadratic, find the solution set. \(3 x(x-4)=3 x^{2}-11 x+4\)

Explain how the zero product property can be used to solve a polynomial equation.

In the mid-nineteenth century, explorers used the boiling point of water to estimate altitude. The boiling temperature of water \(T\) (in \({ }^{\circ} \mathrm{F}\) ) can be approximated by the model \(T=-1.83 a+212,\) where \(a\) is the altitude in thousands of feet. a. Determine the temperature at which water boils at an altitude of \(4000 \mathrm{ft}\). b. Two campers hiking in Colorado boil water for tea. If the water boils at \(193^{\circ} \mathrm{F}\), approximate the altitude of the campers. Give the result to the nearest hundred feet.
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