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Chapter 1: Problem 9

Using the variable \(x\), write each interval using set-builder notation. $$(-\infty,-1]$$

Short Answer

Expert verified
\( \{ x \mid x \leq -1 \} \)

Step by step solution

01

Understand the Interval Notation

The given interval \((-\infty, -1]\) represents all values from negative infinity up to and including -1. The parenthesis \((-\infty\) indicates that -infinity is not included in the set, while the square bracket \([-1]\) indicates that -1 is included.
02

Translate to Set-Builder Notation

In set-builder notation, we express the set by specifying the properties that its elements satisfy. For the interval \((-\infty, -1]\), we want to include all real numbers \(x\) such that \(x\) is less than or equal to -1.
03

Write the Set-Builder Notation

The set-builder notation for the interval \((-\infty, -1]\) can be expressed as: \[ \{ x \mid x \leq -1 \} \] This reads as 'the set of all \(x\) such that \(x\) is less than or equal to -1.'

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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 way to describe a set of numbers, typically a range. It uses parentheses
  • "(" and ")" denote that an endpoint is not included.
  • "[" and "]" are used when an endpoint is included in the interval.
Consider the interval \((-\infty, -1]\). Here, "\((-\infty\)" denotes that the set continues infinitely in the negative direction. The "\([-1]\)" indicates that the number -1 is included. Interval notation provides a clear and compact representation, which is very helpful in mathematics.
Real Numbers
Real numbers encompass all the numbers you would typically encounter in everyday mathematics. They include rational numbers like 1, 2, or -3/4, as well as irrational numbers like the square root of 2 or \pi.

  • Positive and negative integers are part of the real numbers.
  • Decimals and fractions are also real numbers.
  • Irrational numbers, which cannot be written as simple fractions, are included as well.
Real numbers are crucial when discussing intervals. For instance, in the interval \((-\infty, -1]\), every number less than or equal to -1 falls under real numbers. They form the number line that stretches infinitely in both directions, thus connecting directly to intervals and set notations used in mathematics.
Inequality Notation
Inequality notation is another way to express mathematical ranges and relationships between numbers. It uses symbols to show that one number is larger or smaller than another.

  • "<" and ">" show if a number is less than or greater than another respectively.
  • "\(\leq\)" and "\(\geq\)" indicate a number is less than or equal to, or greater than or equal to, another number.
For the interval \((-\infty, -1]\), using inequality notation, you would write: $$x \leq -1$$This clearly communicates that \(x\) includes all real numbers up to and including -1.

Inequality notation is often used alongside interval notation, offering a versatile way to convey mathematical ideas. This notation is especially useful for solving algebraic problems and expressing solutions succinctly.

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