/*! 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 39 Sketch the graph of the inequali... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 39

Sketch the graph of the inequality. $$x<\frac{1}{2}$$

Short Answer

Expert verified
The graph of the inequality is a number line with an open circle at \(\frac{1}{2}\) and shading to the left of this point.

Step by step solution

01

Understand the inequality

The inequality \(x < \frac{1}{2}\) states that the solution includes all \(x\) values that are less than \(\frac{1}{2}\). This does not include \(\frac{1}{2}\) itself since there is no 'or equal to' in the inequality symbol.
02

Draw a number line

Draw a horizontal line to represent the number line. Mark the point where \(x = \frac{1}{2}\) on the line. If the inequality included \(\frac{1}{2}\) itself (i.e., if it had been \(x \leq \frac{1}{2}\)), we would draw a filled dot at this point. However, because our inequality is strictly less than \(\frac{1}{2}\), we draw an open circle at this point.
03

Color the solution range

Since \(x\) is less than but not equal to \(\frac{1}{2}\), this means all values of \(x\) to the left of \(\frac{1}{2}\) are included in the solution. To indicate this, we shade or color the portion of the number line to the left of the open circle.

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 Representation
Graphing inequalities such as \( x < \frac{1}{2} \) on a number line provides a clear visual picture of the solutions it encompasses. A number line is a straight line where each point corresponds to a real number. In representing inequalities, this line becomes a powerful tool to deliver a range or set of numbers precisely.
  • Start by drawing a horizontal line to depict the number line.
  • Choose a reference point鈥攕uch as 0鈥攖o better visualize the scale, and locate \( x = \frac{1}{2} \) accordingly.
  • Mark the position of numbers that are crucial to the inequality, like \( \frac{1}{2} \) in this example.
This clear visual lays the groundwork for the next steps in graphing inequalities and helps in demystifying them.
Inequalities Visualization
When it comes to visualizing inequalities, the aim is to show which side of a particular point on the number line contains the solutions. For the inequality \( x < \frac{1}{2} \), it's important to depict that every number less than \( \frac{1}{2} \) is included in the solution set.To do this:
  • Use an open circle to represent numbers not included in the solution, such as \( \frac{1}{2} \) which is not part of the solution because it's \( x < \frac{1}{2} \), not \( x \leq \frac{1}{2} \).
  • Shade the portion of the number line that extends leftward from \( \frac{1}{2} \), encompassing all numbers less than this value.
  • The shaded area forms a clear indication of all potential solutions that satisfy the inequality.
This visualization turns an abstract inequality into a concrete understanding through graphical representation.
Open and Closed Circles in Graphing
Understanding when to use open or closed circles simplifies the graphing of inequalities. In graphing on a number line, these circles denote whether or not particular points are included in the inequality's solution set.
  • An **open circle** signifies that the number at that point is not included in the solution. For example, \( x < \frac{1}{2} \) is shown with an open circle at \( \frac{1}{2} \), meaning \( \frac{1}{2} \) is not a solution.
  • A **closed circle** means the number is included. If the inequality were \( x \leq \frac{1}{2} \), then \( \frac{1}{2} \) would be represented by a closed circle because it is part of the solution.
These symbols help eliminate ambiguity in inequalities and make it easier to communicate mathematical concepts visually. Understanding these can help you plot inequalities correctly and see at a glance which values satisfy the inequality.

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

Find the slope of the graph of the linear function \(f\). $$f(0)=1, f(4)=-1$$

Sketch the graph of the inequality. $$y \geq 1.5$$

You and two friends each agree to drive \(\frac{1}{3}\) of a 75 -mile trip. Which equation would you use to find the number of miles each of you must drive? A. \(3 x=75\) B. \(\frac{1}{3} x=75\)

Sketch the graph of the inequality. $$\frac{1}{4} x+\frac{1}{2} y<1$$

Check whether \((0,0)\) is a solution. Then sketch the graph of the inequality. $$ x+y>3 $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Mechanics Maths

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.