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Chapter 2: Problem 12

Graph each inequality, and write it using interval notation. \(r<-11\)

Short Answer

Expert verified
\((-\infty, -11)\)

Step by step solution

01

- Understand the inequality

The inequality is given as: \( r < -11 \)This means that the variable \(r\) includes all numbers less than \(-11\).
02

- Draw the number line

Draw a horizontal number line. On this line, identify the point corresponding to \(-11\).
03

- Represent the inequality on the number line

Place an open circle (or hollow dot) at \(-11\) to show that \(-11\) is not included in the solution set. Then shade the number line to the left of \(-11\) to represent all the numbers less than \(-11\).
04

- Write the interval notation

In interval notation, the inequality \( r < -11 \) is represented as:\((-\infty, -11)\)where \( -\infty \) indicates that the set of numbers extends infinitely to the left, and the parenthesis indicates that \(-11\) is not included.

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Key Concepts

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

inequality notation
Inequality notation is a way of expressing the range of values that satisfy an inequality. In the case of the inequality \( r < -11 \), we are expressing that the variable \( r \) takes any value that is less than \(-11\).

This is a strict inequality, meaning that \( r \) does not include the value \(-11\), but anything less. Inequalities can take different forms, such as:
  • \( < \) : less than
  • \( \leq \) : less than or equal to
  • \( > \) : greater than
  • \( \geq \) : greater than or equal to
Understanding these notations is essential as they help us graph and interpret the inequality correctly.
number line
A number line is a visual representation used to show numbers in a straight line, typically drawn horizontally.

To graph inequalities, a number line helps us visualize which values are included in the solution set.
For the inequality \( r < -11 \), follow these steps to represent it on a number line:
  • Draw a horizontal line and mark the point corresponding to \(-11\).
  • Place an open circle (or hollow dot) at \(-11\) to indicate that \(-11\) is not part of the solution.
  • Shade the line extending to the left of \(-11\) to represent all values less than \(-11\).
By using a number line, it's easier to understand the range of values that satisfy the inequality.
interval notation
Interval notation is another compact way of expressing the range of values that an inequality represents.

In interval notation, we use parentheses \(()\) and brackets \([]\) to denote the set of numbers included in the solution:
  • Parentheses \(()\) mean that the endpoint is not included in the interval.
  • Brackets \([]\) mean that the endpoint is included in the interval.
For the inequality \( r < -11 \), the interval notation is \((-fty, -11)\).
This notation tells us:
  • The interval extends infinitely to the left, indicated by \(-\infty\).
  • \(-11\) is an exclusive bound, indicated by the parenthesis.
Using interval notation makes it clear and concise to show which values are part of the solution.

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Most popular questions from this chapter

New York and Ohio are among the states with the most remaining drive-in movie screens. New York has 4 more screens than Ohio, and there are 52 screens total in the two states. How many drive-in movie screens remain in each state? (Data from www. drive-ins.com)

Find the measure of an angle whose supplement measures \(39^{\circ}\) more than twice its complement.

U.S. five-cent coins are made from a combination of nickel and copper. For every \(1 \mathrm{lb}\) of nickel, 3 lb of copper are used. How many pounds of each metal would be needed to make 560 lb of five-cent coins? (Data from The United States Mint.)

Graph the solution set, and write it using interval notation \(-12 \leq \frac{3}{7} x+2 \leq-4\)

Find the measure of an angle whose supplement is eight times its measure. (Hint: If \(x\) represents the measure of the unknown angle, how would we represent its supplement?)
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