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Chapter 0: Problem 11

Use intervals to describe the real numbers satisfying the inequalities. $$x<3$$

Short Answer

Expert verified
\((-fty, 3)\)

Step by step solution

01

Identify the Inequality

The given inequality is \(x < 3\). This states that \(x\) is any number less than 3.
02

Determine the Interval

Since \(x\) can be any number less than 3 but not equal to 3, we represent this using an open interval. The open interval notation for numbers less than 3 is \((-\infty, 3)\).

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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 way of writing subsets of real numbers. It helps in describing a range of values that satisfy particular inequalities.
There are different types of intervals based on whether they include their endpoints:
  • Open intervals: Written as \((a, b)\), do not include the endpoints \(a\) and \(b\)
  • Closed intervals: Written as \([a, b]\), include the endpoints \(a\) and \(b\)
  • Half-open (or half-closed) intervals: Written as \([a, b)\) or \((a, b]\), include only one endpoint while excluding the other
For our exercise, \((-fty, 3)\) is an open interval because it doesn鈥檛 include the endpoint 3.
This means any number less than 3 satisfies the inequality \(x < 3\).
Using interval notation makes it easy to see the range of numbers that fit within a given constraint.
inequality
An inequality is a mathematical statement that compares two values and shows that one value is less than, greater than, or not equal to another value.
Common inequality signs include:
  • \(<\): less than
  • \(\leq\): less than or equal to
  • \(>\): greater than
  • \(\geq\): greater than or equal to
In our example, the inequality is \(x < 3\). This indicates that \(x\) can be any value less than 3.
To solve the inequality, we identify the possible values of \(x\).
Here, since it鈥檚 less than 3, the interval does not include the number 3 itself, leading us to use the open interval notation \((-fty, 3)\).
Inequalities help to understand and visualize the range of values that satisfy certain conditions.
open interval
An open interval \((a, b)\) is a range of numbers between \(a\) and \(b\) that does not include the endpoints \(a\) and \(b\).
For example, in the open interval \((1, 5)\), the set contains numbers greater than 1 and less than 5 but not 1 and 5 themselves.
Open intervals are used in situations where the endpoints cannot be included. This is represented with parentheses rather than square brackets.
In our exercise, the open interval for \(x < 3\) is \((-fty, 3)\):
  • -fty represents no lower limit, indicating all negative and positive numbers less than 3 can be included
  • The absence of the number 3 means 3 is not part of our range
This representation clearly shows which numbers satisfy the inequality while excluding the boundary value.

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Most popular questions from this chapter

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