/*! 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 83 Error Analysis Describe and corr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 83

Error Analysis Describe and correct the error in graphing \(x<5\).

Short Answer

Expert verified
The correct graph will have an open circle at 5 and will be shaded or arrowed to the left.

Step by step solution

01

Understanding the inequality notation

The inequality notation \(x<5\) represents all numbers less than 5. It's important to note that 5 itself is not included in this set, only numbers smaller than 5.
02

Graphing the inequality

To start, a number line is drawn. The number 5 is identified and marked. Because 5 itself is not included in the set of solutions, an open circle should be drawn at 5.
03

Shading the graph

As the inequality is less than 5, meaning all numbers smaller than 5 will fall into the solution of this inequality. Therefore, the area to the left of the marked point (since smaller numbers appear to the left on a number line) should be shaded or arrowed.

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.

Inequality Notation
In mathematical terms, inequality notation describes a range of values that a variable can take. For instance, in the inequality \(x < 5\), the key character is the less-than symbol "<". It dictates that \(x\) can assume any value strictly less than 5.
  • The number 5 is not part of the solution set, given that only numbers smaller than 5 satisfy this inequality.
  • This is crucial to understand for proper representation on a number line or graph.

Inequality notation efficiently expresses a vast collection of numbers. By capturing the idea of a boundary that can't be crossed (in this case, 5), it simplifies communication of mathematical concepts.
Number Line
A number line is a visual representation that helps in understanding and solving inequalities. It is a straight line where each point corresponds to a number. The numbers increase as you move to the right.
  • On this line, specific points can be marked, like the number 5 in our inequality example.
  • Marking these points accurately is essential for great visual aids.

The number line not only allows for a clear depiction of individual numbers but also showcases the entire range of numbers involved in an inequality. This visual guide is particularly useful when learning and communicating mathematical concepts.
Open Circle
When graphing inequalities on a number line, open circles play a significant role. An open circle on a number represents that this number is not included in the solution set covered by the inequality.
  • In \(x < 5\), an open circle is drawn at 5 to indicate that 5 itself is not part of the solution.
  • Open circles neatly convey the idea of excluding the marked number from the set.

Understanding the use of open circles helps ensure the accurate representation of inequalities on a number line. This simple symbol effectively denotes the boundaries of numerical values in the solution set.
Shading Graph
Shading a graph is an integral step in visually representing solutions to inequalities on a number line. Shading shows which part of the line is included in the solution.
  • For \(x < 5\), the shaded region lies to the left of the open circle at 5, representing all numbers less than 5.
  • The leftward shading signifies that all smaller numbers are included in the set.

Shading provides a clear visual signal of where solutions exist on a number line. It simplifies understanding by directly illustrating which areas numbers fall into the inequality solution, thereby enhancing comprehension and communication of mathematical concepts.

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

Solving an Inequality Does dividing each side of an inequality by 5 yield the same result as multiplying each side by \(\frac{1}{5}\) ? Give an example.

Travel Time On the first part of a 225-mile automobile trip, you averaged 55 miles per hour. On the last part of the trip, you averaged 48 miles per hour because of increased traffic congestion. The total trip took 4 hours and 15 minutes. Find the travel time for each part of the trip.

Determine whether the statement is true or false. Justify your answer. $$x

Solve and graph the inequality. $$6+\frac{2 x}{3}

Solve and graph the inequality. $$-10 x<40$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

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.