/*! 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 13 Graph the inequality. $$x<2... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 13

Graph the inequality. $$x<2$$

Short Answer

Expert verified
Use an open circle at 2 and shade all points to the left on the number line.

Step by step solution

01

Understand the Inequality

An inequality is a mathematical sentence that shows the relationship between quantities that are not equivalent. In this case, we have the inequality \(x < 2\), which indicates all values that are less than 2.
02

Identify the Boundary Point

The inequality \(x < 2\) has a boundary point at \(x = 2\). This is the point you will use as a reference when graphing.
03

Choosing the Correct Symbol

Since the inequality is less than \(x < 2\) and does not include 2, we will use an open circle on the number line at \(x = 2\). Open circles indicate the value is not included.
04

Determine the Shaded Region

Because the inequality is \(x < 2\), shade the region to the left of \(x = 2\) on the number line. This represents all values less than 2.
05

Graph the Solution

Draw a number line, mark the point \(x = 2\) with an open circle, and shade the region to the left. This shows all the points on the number line that satisfy the inequality \(x < 2\).

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.

Graphing Inequalities
Graphing inequalities is a visual way to represent mathematical expressions that show the relationship between quantities that aren't equal. Let's take the inequality \(x < 2\) as an example. Here are the steps to graph an inequality on a number line:
  • Identify the inequality symbol (such as <, >, 鈮�, or 鈮�) to understand the relationship. For \(x < 2\), it means the values of \(x\) are less than 2.
  • Choose how to represent this on a number line. A number line is a straight horizontal line with numbers placed at intervals.
  • Use an open circle or a closed dot based on whether the number is included in the inequality or not. For example, with \(x < 2\), we do not include 2, so we use an open circle at \(x = 2\).
With these basics, you can effectively translate any simple inequality into a graphical format using a number line.
Number Line Representation
A number line representation is a simple way to show the range of possible solutions for an inequality. When plotting these solutions for an inequality like \(x < 2\), follow these steps:First, draw a horizontal line and mark numbers at regular intervals on it. This creates a visual reference point.
  • Locate the critical value related to the inequality, which is 2 in this case.
  • Place an open circle directly above the 2 on this line. The open circle signifies that 2 itself is not part of the set of solutions.
  • Shade the area or draw an arrow extending to the left of the open circle. This shaded region or arrow points to all the numbers that satisfy the condition \(x < 2\).
With these simple steps, the number line becomes a powerful tool to visually display inequality solutions.
Boundary Points
Boundary points are crucial when graphing inequalities because they define the edges of your solution set on a number line. In \(x < 2\), the boundary point is at \(x = 2\):
  • A boundary point is where an inequality changes from true to false, or vice versa. It acts as a reference that shows the threshold for acceptable values.
  • Determine whether to include the boundary point. Since \(x < 2\) excludes 2, we use an open circle to mark this key point. This open circle communicates that 2 itself is not part of any solution to \(x < 2\).
  • Knowing where to place the boundary point helps ensure the inequality is represented accurately.
Understanding boundary points helps clarify why specific parts of the number line are shaded or left out, ensuring a correct and complete graph of an 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 partial fraction decomposition of the rational function. $$\frac{x-12}{x^{2}-4 x}$$

Find the partial fraction decomposition of the rational function. $$\frac{x+6}{x(x+3)}$$

Solving a Linear System as a Matrix Equation Solve the system of equations by converting to a matrix equation and using the inverse of the coefficient matrix, as in Example \(6 .\) $$\left\\{\begin{aligned}-2 y+2 z &=12 \\\3 x+y+3 z &=-2 \\\x-2 y+3 z &=8\end{aligned}\right.$$

Solving a Linear System Solve the system of equations by converting to a matrix equation. Use a graphing calculator to perform the necessary matrix operations, as in Example 7. $$\left\\{\begin{array}{l}x+y+z+w=15 \\\x-y+z-w=5 \\\x+2 y+3 z+4 w=26 \\\x-2 y+3 z-4 w=2\end{array}\right.$$

Find the partial fraction decomposition of the rational function. $$\frac{2 x+1}{x^{2}+x-2}$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Decision 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.