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Chapter 1: Problem 56

Graph the solution set and give the interval notation equivalent. \(\quad-40

Short Answer

Expert verified
Interval notation: \((-40, 0)\).

Step by step solution

01

Identify the Inequality Range

The given inequality is \(-40 < x < 0\). Our task is to identify the range of values for \(x\) which are all greater than \(-40\) and less than \(0\).
02

Understanding Interval Notation

In interval notation, \(-40 < x < 0\) is written as \((-40, 0)\), where the parentheses indicate that \(-40\) and \(0\) are not included in the solution set.
03

Graph the Solution Set

To graph the solution set on a number line, draw a line and place an open circle (or a dot) at \(-40\) and at \(0\) to indicate that these values are not included. Then, shade the region between \(-40\) and \(0\) to indicate that all numbers in this range are included.

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Key Concepts

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

graphing inequalities
When we graph inequalities, we visually represent solutions on a number line. This helps us quickly see the range of values that satisfy the inequality. The inequality \(-40 < x < 0\) means that \(x\) is any number greater than \(-40\) and less than \(0\).
To graph this on a number line, follow these steps:
  • Draw a horizontal line, which we call the number line.
  • Mark the points \(-40\) and \(0\) on this line.
  • Place open circles or dots at these points to show that \(-40\) and \(0\) are not part of the solution set.
  • Shade the section of the number line between these two points to show that all numbers in this interval are included.
This graphical representation efficiently communicates which values satisfy the original inequality.
interval notation
Interval notation is a concise way to describe a range of values. It clearly indicates the start and end points of an interval, as well as whether these endpoints are included or excluded. For the inequality \(-40 < x < 0\), we want to cover values greater than \(-40\) and less than \(0\).
The interval notation for this is \((-40, 0)\), where:
  • The first number, \(-40\), indicates the lower bound of the interval, but not included because of the \(<\) symbol.
  • The second number, \(0\), represents the upper bound, also not included, because the inequality does not include equality (i.e., the \(<\) symbol).
  • Parentheses \( \left( \right) \) denote that the endpoints are not included. If an endpoint were included, we would use square brackets \( \left[ \right] \).
Interval notation is especially useful in mathematics for its simplicity and clarity, particularly when expressing solutions of inequalities.
number line representation
Number line representation is a simple, visual method to demonstrate the range of solutions for an inequality. Consider the inequality \(-40 < x < 0\). This tells us that \(x\) is a value strictly between \(-40\) and \(0\).
To represent this on a number line:
  • Draw a straight, horizontal line. This is your number line.
  • Plot relevant points (in this case, \(-40\) and \(0\)) on the number line.
  • Use open circles at \(-40\) and \(0\) to show these numbers are not included in the solution set 鈥� contrast this with closed circles, which would indicate inclusion.
  • Finally, shade or darken the line segment between these two points. This shading indicates the range of numbers, illustrating where values of \(x\) lie.
By using a number line, you pinpoint exactly where an inequality's solutions fall, making it easier to understand and interpret the problem's conditions.

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