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Chapter 1: Problem 41

Express the interval in terms of inequalities, and then graph the interval. $$(-3,0)$$

Short Answer

Expert verified
The interval (-3,0) as an inequality is -3 < x < 0.

Step by step solution

01

Understanding the Interval Notation

The interval (-3,0) is in interval notation where "(" and ")" denote that the endpoints of the interval are not included. This interval contains all numbers between -3 and 0, but does not include -3 or 0 themselves.
02

Convert Interval to Inequality

To express the interval (-3,0) as an inequality, observe that the values range from greater than -3 to less than 0. Thus, the inequality is -3 < x < 0. Here, "x" represents all numbers in this range.
03

Graph the Inequality on a Number Line

First, draw a horizontal number line. Locate the points -3 and 0, and draw open circles over these points to indicate they are not included in the interval. Shade the line between these two points to represent all numbers between -3 and 0.

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Key Concepts

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

Understanding Inequalities
Inequalities are a way to express mathematical statements where two values are compared using symbols like '<' (less than), '>' (greater than), '鈮�' (less than or equal to), and '鈮�' (greater than or equal to). They are very useful in representing ranges of values, especially when certain numbers are not included in that range. Consider the inequality \(-3 < x < 0\). This means any value of \(x\) is greater than -3 and less than 0.
  • \(<\) and \(>\) indicate that endpoints are not part of the solution.
  • \(\leq\) and \(\geq\) include endpoints.
When you convert interval notation, like \((-3, 0)\), into an inequality, you clearly specify that no endpoint is included by using '<' or '>' rather than including them.
Using a Number Line for Inequalities
A number line is a visual representation of numbers in a straight line, which helps us understand and visualize inequalities with ease. When graphing inequalities such as \(-3 < x < 0\), the number line becomes a tool to visually show which numbers satisfy the inequality.
  • First, plot the key points, here -3 and 0, on the number line.
  • Use open circles on these points to signify they are not included in the solutions, matching our '<' inequality notation.
  • Shading the area between these open circles represents all solutions \(x\) can take.
This way, the number line provides a clear and visual statement of the inequality.
Exploring Open Intervals
Open intervals are specific types of intervals in mathematics where the endpoints are not part of the interval. Notation such as \((-3, 0)\) signifies an open interval. Understanding these intervals is crucial because they help define ranges that exclude their boundary points. Here's what makes open intervals unique:
  • The parentheses \("(" and ")"\) indicate the endpoints are not included.
  • In contrast, 'closed intervals' use brackets \("[" and "]"\) to include endpoints.
  • Open intervals align with inequalities that do not saturate endpoints, using \(<\) and \(>\).
Thus, open intervals efficiently describe continuous sections of a number line without including specified limits.

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