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Chapter 1: Problem 2

Using interval notation, write each set. Then graph it on a number line. $$\\{x | x \geq-3\\}$$

Short Answer

Expert verified
Interval notation: \([-3, \infty)\); Solid dot at -3, line to the right.

Step by step solution

01

Understand the Set Notation

The given set is \( \{x | x \geq -3\} \), which means it includes all numbers \( x \) such that \( x \) is greater than or equal to \(-3\).
02

Convert to Interval Notation

In interval notation, the set \( \{x | x \geq -3\} \) is written as \([-3, \infty)\). The bracket \([\) indicates that -3 is included in the set, and the parenthesis \()\) after infinity indicates that infinity is not included.
03

Graph on a Number Line

To graph \([-3, \infty)\) on a number line, draw a solid dot at -3 to indicate that -3 is included. Then draw a line extending to the right from the dot, indicating all numbers greater than -3.

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Key Concepts

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

Set Notation
Set notation is a simple way to express a collection of elements, often numbers, based on specific conditions. It's like stating the rules for membership in a club. For instance, in the set notation \( \{x | x \geq -3\} \), the vertical bar \(|\) is like saying "such that." This means we are talking about all numbers \(x\) such that they satisfy the condition \(x \geq -3\).
  • The curly braces \(\{ \} \) symbolize that we are describing a set of values.
  • The statement after the bar specifies the condition, which is often expressed as an inequality.
This notation is essential in mathematics because it provides a clear, concise way to describe a potentially infinite collection of numbers that meet specific criteria.
Number Line
A number line is a visual representation of numbers along a horizontal line, which can help make abstract concepts more tangible. It's like a ruler that extends in both directions forever.
  • Each point on the line corresponds to a unique real number.
  • The center of the number line is typically at zero, with positive numbers extending to the right and negative numbers extending to the left.
  • Infinity points are depicted by arrows pointing beyond the number ends.
To graph an interval like \([-3, \infty)\), you place a solid dot above the number \(-3\) to show inclusion. Then, extend a line to the right, highlighting all numbers larger than \(-3\) because they are included in the set. Drawing intervals on a number line helps visualize solutions to inequalities, showcasing the range of values that meet the condition.
Inequalities
Inequalities are mathematical expressions that describe the relationship between two values in terms of their size. They explain if one value is smaller, greater, or equal to another.
  • The symbol \( \geq \) means "greater than or equal to."
  • \( \leq \) stands for "less than or equal to."
  • \( > \) and \( < \) denote strict "greater than" and "less than," respectively, excluding equality.
For instance, \( x \geq -3 \) means \(x\) can be any number from \(-3\) onwards. Solutions of inequalities can be graphically shown as intervals on the number line or through interval notation. Using inequalities alongside other forms of notation helps define and solve mathematical problems effectively, providing clarity in the solution range.

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Most popular questions from this chapter

The following table lists Square's daily transactions \(y\) in millions of dollars, \(x\) months after March 2011 . (Square can be used to accept credit cards on your iPhone.) $$\begin{array}{c|c} x & y \\ \hline 0 & 1.0 \\ 2 & 2.0 \\ 7 & 5.5 \\ 12 & 11.0 \end{array}$$ (a) Using ordered pairs, write a function \(T\) that gives the daily transactions in millions of dollars during each month. Interpret the first ordered pair. (b) Repeat part (a) using a diagram. (c) Identify the domain and range of \(T\).

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=3 x-4, x=-2$$

Given an equation having \(x\) and \(y\) as variables, explain how to determine the \(x\) - and \(y\) -intercepts.

Using the variable \(x\), write each interval using set-builder notation. $$[2,7)$$

Find \(f(a), f(b+1),\) and \(f(3 x)\) for the given \(f(x)\) $$f(x)=5 x$$
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