/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 23 Use graphs to find each set. $... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 23

Use graphs to find each set. $$[3, \infty) \cap(6, \infty)$$

Short Answer

Expert verified
The intersection of sets \([3, \infty)\) and \((6, \infty)\) is \((6, \infty)\)

Step by step solution

01

Graph the first interval

Graph the interval \([3, \infty)\) on a number line. This interval includes all numbers, starting from (including) 3 and extending infinitely in the positive direction. Mark a solid dot on 3 and draw a line extending off to the right to symbolize going to infinity.
02

Graph the second interval

Now, graph the interval \((6, \infty)\) on the same number line. This interval includes all numbers that are strictly greater than 6 and extend infinitely towards the positive direction. Mark an open circle on 6 and draw a line extending off to the right, again symbolizing that it's going to infinity.
03

Find the intersection

The intersection of the two intervals is represented by the section of the number line that both intervals have in common. Here, it can be seen that the common section starts from 6 (excluding 6) and extends to infinity. Therefore, the intersection is \((6, \infty)\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Number Line
Understanding a number line is essential when working with intervals. It is a straight, horizontal line with numbers placed at equal intervals along its length. Each point on the line corresponds to a real number.

When dealing with intervals, the number line helps us visually represent these sets of numbers. For instance:
  • A filled-in dot is used to indicate a number that is included in the interval (closed interval).
  • An open circle is used for numbers not included in the interval (open interval).
This graphical representation makes it easier to see where numbers or intervals start and end.
Intersection of Intervals
The intersection of intervals refers to the set of numbers that belong to both intervals simultaneously. We represent this graphically by plotting both intervals on a number line and identifying the overlapping section.

Consider two intervals,
  • ":[3, \infty)" includes 3 and all numbers greater than 3.
  • "(6, \infty)" includes only numbers greater than 6.
To find the intersection:
  • Plot each interval on the number line.
  • Identify where their covered areas overlap.
In this case, the intersection is \((6, \infty)\), since both intervals only share numbers greater than 6.
Graphing Intervals
Graphing intervals is a simple way to visualize the range of numbers included in a set. To graph an interval:
  • Identify the start and end of the interval.
  • Use a solid line for continuous ranges.
  • Mark dots or circles at the relevant points.
For example, the interval \([3, \infty)\) is represented by a line starting at 3 with a solid dot, moving right infinitely. An interval like \((6, \infty)\) will start with an open circle at 6, extending rightward.

By visualizing intervals, it's easier to see intersections and understand how they interact on a number line. Graphing offers an intuitive approach to solving problems involving intervals.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

If a coin is tossed 100 times, we would expect approximately 50 of the outcomes to be heads. It can be demonstrated that a coin is unfair if \(h,\) the number of outcomes that result in heads, satisfies \(\left|\frac{h-50}{5}\right| \geq 1.645 .\) Describe the number of outcomes that determine an unfair coin that is tossed 100 times.

Factor the numerator and the denominator. Then simplify by dividing out the common factor in the numerator and the denominator. $$\frac{x^{2}+6 x+5}{x^{2}-25}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The cube root of \(-8\) is not a real number.

Solve each equation. $$\frac{x-1}{x-2}+\frac{x}{x-3}=\frac{1}{x^{2}-5 x+6}$$

Explain how to determine the restrictions on the variable for the equation $$\frac{3}{x+5}+\frac{4}{x-2}=\frac{7}{x^{2}+3 x-6}$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.