/*! 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 21 Graph each linear inequality. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 21

Graph each linear inequality. $$2 x+3 y>-6$$

Short Answer

Expert verified
Rewrite as \(y > -\frac{2}{3}x - 2\), graph the boundary line with slope \(-\frac{2}{3}\) and y-intercept -2, and shade above the line.

Step by step solution

01

Rewrite the Inequality in Slope-Intercept Form

To graph the inequality, first rewrite it in the form of a linear equation. Start by isolating the y-term on one side. Given inequality is \(2x + 3y > -6\). Subtract 2x from both sides to get \(3y > -2x - 6\). Now divide by 3: \(y > -\frac{2}{3}x - 2\).
02

Graph the Boundary Line

Graph the equation \(y = -\frac{2}{3}x - 2\) as if it were an equality. Use a dashed line because the symbol '>' indicates that points on the line are not included in the solution set. Find the y-intercept at -2 and use the slope \(-\frac{2}{3}\) to plot additional points.
03

Shade the Appropriate Side

To determine which side to shade, pick a test point not on the line (0, 0) is an easy choice. Substitute (0, 0) into the inequality: \(2(0) + 3(0) > -6\) which simplifies to \(0 > -6\), a true statement. Therefore, shade the region that includes (0, 0). This is the area above the dashed line.

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.

Slope-Intercept Form
Understanding the slope-intercept form of a linear inequality is key to graphing it effectively. The slope-intercept form of a linear equation is written as:
\[y = mx + b\]
Here, \(m\) represents the slope of the line, and \(b\) represents the y-intercept, where the line crosses the y-axis.
To convert the given inequality \(2x + 3y > -6\) to this form, follow these steps:
  • First, isolate the \(y\)-term: subtract \(2x\) from both sides to get \(3y > -2x - 6\).
  • Then divide every term by 3 to solve for \(y\): \(y > -\frac{2}{3}x - 2\).
Now the inequality is in slope-intercept form, where:
- The slope \(m\) is \(-\frac{2}{3}\).
- The y-intercept \(b\) is \(-2\).
This form makes it easier to plot the inequality on a graph.
Boundary Line
The boundary line is critical when graphing linear inequalities. It shows the points that are not included in the solution set.
To graph the boundary line for the inequality \(2x + 3y > -6\), follow these steps:
  • Replace the inequality symbol with an equality symbol to get the equation \(y = -\frac{2}{3}x - 2\).
  • Plot the y-intercept \(-2\) on the y-axis.
  • Use the slope \(-\frac{2}{3}\) to find another point. From the y-intercept \(-2\), move down 2 units and right 3 units (since the slope is negative).
  • Draw a dashed line through these points. The line is dashed because the inequality does not include the points on the line (i.e., \( > \) or \( < \)).
The dashed boundary line represents the edge of our solution set but does not include the points on the line itself.
Shading Regions
Shading is the final step in graphing a linear inequality and indicates the solutions to the inequality.
To determine which side of the boundary line to shade:
  • Choose a test point that is not on the boundary line. The origin (0, 0) is often a good choice because it simplifies calculations.
  • Substitute the test point into the inequality: \(2(0) + 3(0) > -6\) simplifies to \(0 > -6\).
  • Since this statement is true, the solution includes the region that contains (0, 0). Shade this area.
This shading process demonstrates that all points in the shaded region satisfy the inequality \(2x + 3y > -6\). Remember, shading above the dashed line means that any point in this region will make the inequality true.

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

Describe what the graph of each linear equation will look like in the coordinate plane. (Hint: Rewrite the equation if necessary so that it is in a more recognizable form.) $$ 3 x=y-9 $$

Graph each linear equation. \(-3 y=15\)

Find each quotient. $$ \frac{5-(-7)}{-4-(-1)} $$

Graph each linear equation. \(y=-6 x\)

Find the \(x\) -intercept and the \(y\) -intercept for the graph of each equation. $$ y=2.5 $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

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.