### Graphing Systems of Inequalities The image presents a system of inequalities along with a graph that is meant to represent the solutions to the system. The system given is: 1. \(-2x + y \leq -4\) 2. \(-2x + 4y \geq -8\) #### Explanation of the Graph The graph shown is on a coordinate plane ranging from \(-7\) to \(7\) on both the x-axis and the y-axis. The grid lines are marked at integer values to help plot points and draw the boundary lines for the inequalities. - **First Inequality \(-2x + y \leq -4\):** - To graph, convert the inequality to the equation of a line: \(y = 2x - 4\). - Plot the line with \(y\)-intercept \(-4\) and slope \(2\) (rise over run is 2 units up for each unit right). - Shade the region below the line since the inequality is \(\leq\). - **Second Inequality \(-2x + 4y \geq -8\):** - Convert to the line equation: \(4y = 2x - 8\) or \(y = \frac{1}{2}x - 2\). - Plot the line with \(y\)-intercept \(-2\) and slope \(\frac{1}{2}\) (rise 1 unit for every 2 units right). - Shade the region above the line since the inequality is \(\geq\). #### Intersection of the Regions The solution to the system of inequalities is the region where the shaded areas overlap. This represents all the pairs \((x, y)\) that satisfy both inequalities simultaneously. Feel free to experiment with different values of \(x\) to verify their positions relative to the boundary lines and identify suitable \(y\) values that satisfy each inequality.
### Graphing Systems of Inequalities The image presents a system of inequalities along with a graph that is meant to represent the solutions to the system. The system given is: 1. \(-2x + y \leq -4\) 2. \(-2x + 4y \geq -8\) #### Explanation of the Graph The graph shown is on a coordinate plane ranging from \(-7\) to \(7\) on both the x-axis and the y-axis. The grid lines are marked at integer values to help plot points and draw the boundary lines for the inequalities. - **First Inequality \(-2x + y \leq -4\):** - To graph, convert the inequality to the equation of a line: \(y = 2x - 4\). - Plot the line with \(y\)-intercept \(-4\) and slope \(2\) (rise over run is 2 units up for each unit right). - Shade the region below the line since the inequality is \(\leq\). - **Second Inequality \(-2x + 4y \geq -8\):** - Convert to the line equation: \(4y = 2x - 8\) or \(y = \frac{1}{2}x - 2\). - Plot the line with \(y\)-intercept \(-2\) and slope \(\frac{1}{2}\) (rise 1 unit for every 2 units right). - Shade the region above the line since the inequality is \(\geq\). #### Intersection of the Regions The solution to the system of inequalities is the region where the shaded areas overlap. This represents all the pairs \((x, y)\) that satisfy both inequalities simultaneously. Feel free to experiment with different values of \(x\) to verify their positions relative to the boundary lines and identify suitable \(y\) values that satisfy each inequality.
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
Transcribed Image Text:### Graphing Systems of Inequalities
The image presents a system of inequalities along with a graph that is meant to represent the solutions to the system. The system given is:
1. \(-2x + y \leq -4\)
2. \(-2x + 4y \geq -8\)
#### Explanation of the Graph
The graph shown is on a coordinate plane ranging from \(-7\) to \(7\) on both the x-axis and the y-axis. The grid lines are marked at integer values to help plot points and draw the boundary lines for the inequalities.
- **First Inequality \(-2x + y \leq -4\):**
- To graph, convert the inequality to the equation of a line: \(y = 2x - 4\).
- Plot the line with \(y\)-intercept \(-4\) and slope \(2\) (rise over run is 2 units up for each unit right).
- Shade the region below the line since the inequality is \(\leq\).
- **Second Inequality \(-2x + 4y \geq -8\):**
- Convert to the line equation: \(4y = 2x - 8\) or \(y = \frac{1}{2}x - 2\).
- Plot the line with \(y\)-intercept \(-2\) and slope \(\frac{1}{2}\) (rise 1 unit for every 2 units right).
- Shade the region above the line since the inequality is \(\geq\).
#### Intersection of the Regions
The solution to the system of inequalities is the region where the shaded areas overlap. This represents all the pairs \((x, y)\) that satisfy both inequalities simultaneously.
Feel free to experiment with different values of \(x\) to verify their positions relative to the boundary lines and identify suitable \(y\) values that satisfy each inequality.
Expert Solution
Step 1
To graph the following inequalities
Step by step
Solved in 3 steps with 2 images
Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education