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Chapter 2: Problem 111

Determine the inequality given the answers expressed in interval notation. $$ (-\infty, 3) $$

Short Answer

Expert verified
The inequality is \(x < 3\).

Step by step solution

01

Understand Interval Notation

The interval \((-fty, 3)\) represents all numbers less than 3. It includes all real numbers from negative infinity up to (but not including) 3. The negative infinity symbol \(-fty\) signifies that there is no lower bound, just an upper limit of 3, which is not included in the interval.
02

Translate to Inequality Notation

The interval \((-fty, 3)\) in inequality form is expressed as \(x < 3\). This means any value for \(x\) that is less than 3 satisfies the inequality.

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Key Concepts

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

Interval Notation
Interval notation is a mathematical method used to describe a range of numbers along the number line. It's a compact way to write inequalities and efficiently communicates the idea of a set of numbers. For example, the interval
  • \((-fty, 3)\) means all real numbers less than 3.
When you see parentheses like
  • \(\{3\}\) or \((3)\), it indicates that the number 3 is not included in the set.
Conversely, the symbols
  • \([3]\) or \((3]\) would imply that 3 is included.
Interval notation often uses the infinity symbol (fty) to imply no upper or lower boundary.
For example,
  • \((-fty, a)\) means the set of all numbers less than \(a\),
  • \([a, \infty)\) means all numbers greater than or equal to \(a\).
It's important to note that infinity is a concept, not a number, so it can never be included in an interval.
Inequality Notation
Inequality notation is a way to represent a range of values that satisfy a particular condition. It involves symbols like
  • \(<\),
  • \(>\),
  • \(\leq\) (less than or equal to), and
  • \(\geq\) (greater than or equal to)
to show the relation between quantities. The inequality \(x < 3\) means any number \(x\) that is less than 3 is a part of the solution set.
To translate from interval notation to inequality notation:
  • Identify the endpoints in the interval.
  • Determine if they are included (use \(\leq\) or \(\geq\)) or not included (use \(<\) or \(>\)) based on parentheses or brackets.
For example, in \((-\infty, 3)\), \(-\infty\) suggests there is no lower bound, so you only look at 3 which is not included. Thus, we use \(<\) to show that our values range up to but do not touch 3. This translates directly to \(x < 3\).
Real Numbers
The real numbers include every possible number along the number line from negative infinity to positive infinity. They encompass
  • everything from decimals,
  • fractions, and
  • whole numbers,
forming a continuous line with no gaps.
Real numbers can be broken into several categories:
  • Rational numbers, such as 1/2 or -3, which can be expressed as a fraction of two integers
  • Irrational numbers, like \(\pi\) or \(\sqrt{2}\), which cannot be exactly expressed as fractions
When dealing with intervals like \((-\infty, 3)\), it's understood that all numbers less than 3, covering all decimal and fractional subvalues, are included. Real numbers are crucial because they allow us to represent a continuous set of numbers, rather than just a finite selection of integers.

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Most popular questions from this chapter

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -10 \leq 3 x-1 \leq-4 $$

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Graph all solutions on a number line and provide the corresponding interval notation. $$ -12 \leq x<32 $$

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