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Chapter 2: Problem 12

Write each inequality in interval notation, and graph the interval. \(r<-11\)

Short Answer

Expert verified
Interval notation: \((-\infty, -11)\), Open circle at -11 shading left.

Step by step solution

01

- Understand the inequality

The inequality given is \(r < -11 \). This means that r can be any number that is less than -11.
02

- Write the inequality in interval notation

In interval notation, \(r < -11\) is written as \((-\infty, -11)\). The parenthesis '(' indicates that -11 is not included in the interval, and '-\infty' means it goes to negative infinity.
03

- Graph the interval

To graph \((-\infty, -11)\), draw a number line and shade everything to the left of -11. Use an open circle at -11 to show that -11 is not included in the interval. This visually represents all values less than -11.

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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 shorthand way of expressing sets of numbers. For instance, when we want to describe all numbers less than -11, we use interval notation to clearly convey this. In the example given, \(r < -11 \) is written as \((-fty, -11)\). Here鈥檚 the breakdown:
  • The '(' means that the interval starts at -鈭� and does not include -11.
  • '-鈭�' indicates that the numbers go infinitely in the negative direction.
  • The '-11' with ')' shows that -11 is the endpoint but is not included.
This notation helps students quickly identify the range of numbers involved.
graphing inequalities
Graphing inequalities provides a visual representation of the range of values that satisfy the inequality. Consider the inequality \(r < -11\). To graph this:
  • Draw a horizontal number line.
  • Identify the point -11 on this line.
  • Use an open circle at -11 to show that -11 is not included in the interval.
  • Shade the area to the left of -11, indicating all numbers less than -11 are part of the solution.
This shading visually represents the solution on the number line, making it easier to understand and interpret the range.
number line
A number line is a tool used in mathematics to represent numbers in a linear format. It includes both positive and negative numbers, with 0 at the center. In the context of inequalities:
  • To represent numbers less than a certain value, shade the region to the left of that number.
  • Use open or closed circles to indicate whether the endpoint is included or not. For \(r < -11 \), we use an open circle at -11.
Number lines provide a simple and effective way to visualize solutions to inequalities, making abstract concepts more concrete for learners.

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