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Chapter 1: Problem 106

Translate to an inequality. The distance from \(t\) to 0 is at least \(20 .\)

Short Answer

Expert verified
\( |t| \geq 20 \)

Step by step solution

01

Identify Key Information

The exercise states that the distance from a variable, denoted as \( t \), to 0 must be at least 20.
02

Understand the Concept of Distance

Distance in a mathematical context represents the absolute value. The absolute value of \( t \) from 0 is represented as \( |t| \).
03

Set Up the Inequality

Since the distance must be at least 20, it is expressed as: \( |t| \geq 20 \). The symbol \( \geq \) means 'greater than or equal to'.

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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 an essential concept in mathematics, especially when dealing with inequalities. Think of absolute value as the distance a number is from zero on the number line, without considering direction. For any given number, whether it is positive or negative, the absolute value is always positive.
For instance:
  • The absolute value of 5 is written as \(|5|\), and equals 5.
  • The absolute value of -5 is written as \(|-5|\), and also equals 5.
In our exercise, we have a variable, \(|t|\), and the distance from 0 must be at least 20. So, the absolute value inequality becomes \[|t| \geq 20.\]
This means that \(t\) can be any number 20 units away from 0 or more, including 20 and -20.
Mathematical Distance
Distance in mathematics often refers to how far two points are from each other on a number line. In our case, we are looking at the distance of a variable, \(t\), from the number 0. This is expressed using absolute value.
The concept of distance is important in understanding inequalities. In our problem, the statement 'the distance from \(t\) to 0 is at least 20' tells us that \(|t| \geq 20\).
To break it down:
  • 'Distance' translates to absolute value, so we write \(|t|\).
  • 'At least 20' means the value must be 20 or more, not less. Hence, we use the symbol \(\geq\), which means 'greater than or equal to'.
Put together, we have \(|t| \geq 20\). This indicates that \(t\) can be any number where the distance from 0 is exactly 20 or more.
Variable
A variable in mathematics is a symbol, often a letter, that represents an unknown value. Variables are used to express equations and inequalities more generally.
In our exercise, \(t\) is the variable. We do not know its exact value, but we do know something about it: its distance from 0. The inequality \[|t| \geq 20\] tells us that the value of \(t\) is at least 20 units away from zero.
To better understand variables:
  • Variables can stand for any number within a given set. For example, \(t\) might be any real number in this exercise.
  • They allow mathematicians to create general relationships and rules that apply to many situations.
In practice, if we solve the inequality \[|t| \geq 20\], \(t\) could be 20, -20, or any number greater than or equal to 20 or less than or equal to -20. So \(t\) represents a whole range of possible numbers.

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