Graphing Systems of Linear Inequalities: Unveiling the Overlap

Algebra 2 Grades High School 32:03 Video
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Lesson Description

Master the art of graphing systems of linear inequalities and identifying solution regions where inequalities overlap. This lesson reinforces linear inequality graphing and introduces the concept of systems.

Video Resource

Linear Inequalities Systems Graph

Kevinmathscience

Duration: 32:03
Watch on YouTube

Key Concepts

  • Linear Inequalities
  • Systems of Inequalities
  • Graphing Linear Inequalities
  • Solution Region (Overlap)
  • Slope-intercept form (y = mx + b)
  • Dashed vs. Solid Lines

Learning Objectives

  • Graph linear inequalities on the coordinate plane.
  • Identify the solution region of a system of linear inequalities.
  • Determine whether a line should be dashed or solid based on the inequality symbol.
  • Rewrite linear inequalities into slope-intercept form.
  • Interpret the meaning of the solution region in the context of the inequalities.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concept of graphing single linear inequalities. Briefly discuss the meaning of inequality symbols (>, <, ≥, ≤) and how they relate to shading above or below the line. Preview the concept of a 'system' as having multiple inequalities.
  • Video Presentation (15 mins)
    Play the Kevinmathscience video 'Linear Inequalities Systems Graph'. Encourage students to take notes on the steps involved in graphing systems of inequalities. Emphasize the importance of identifying the overlap region.
  • Guided Practice (20 mins)
    Work through several examples similar to those in the video, demonstrating the process step-by-step. Emphasize rewriting inequalities in slope-intercept form, identifying the slope and y-intercept, determining whether to use a dashed or solid line, and shading the correct region. Focus on identifying the overlapping region as the solution set.
  • Independent Practice (15 mins)
    Provide students with a worksheet containing a variety of systems of linear inequalities to graph. Encourage students to work individually or in pairs, providing assistance as needed. Review answers as a class.
  • Wrap-up and Assessment (5 mins)
    Summarize the key concepts of graphing systems of linear inequalities. Administer a short multiple choice or fill-in-the-blank quiz to assess student understanding.

Interactive Exercises

  • Desmos Activity
    Use Desmos to create an interactive activity where students can graph systems of inequalities and explore the solution regions in real time. Students can adjust the inequalities and observe how the solution region changes.
  • Error Analysis
    Present students with graphs of systems of inequalities that contain common errors (e.g., incorrect shading, dashed lines used instead of solid lines). Ask students to identify and correct the errors.

Discussion Questions

  • What does the overlapping region represent in the graph of a system of inequalities?
  • How does the inequality symbol determine whether you use a dashed or solid line?
  • Why is it helpful to rewrite inequalities in slope-intercept form before graphing?
  • Can a system of linear inequalities have no solution? If so, what would that look like graphically?

Skills Developed

  • Graphing Linear Equations and Inequalities
  • Solving Systems of Equations
  • Algebraic Manipulation
  • Problem-Solving
  • Visual Representation of Mathematical Concepts

Multiple Choice Questions

Question 1:

Which of the following is the first step in graphing a system of linear inequalities?

Correct Answer: Rewrite the inequalities in slope-intercept form.

Question 2:

What does a dashed line indicate when graphing an inequality?

Correct Answer: The points on the line are not included in the solution.

Question 3:

The solution to a system of linear inequalities is represented by:

Correct Answer: The overlapping region of all shaded areas.

Question 4:

Which inequality symbol indicates shading above the line?

Correct Answer: >

Question 5:

What is the slope-intercept form of a linear equation?

Correct Answer: y = mx + b

Question 6:

When do you flip the inequality sign?

Correct Answer: When multiplying or dividing by a negative number.

Question 7:

What does it mean if a system of linear inequalities has no solution?

Correct Answer: The lines are perpendicular.

Question 8:

Which of the following points is a solution to the system: y > x + 1 and y < -x + 3?

Correct Answer: (0, 2)

Question 9:

What type of line should you draw for the inequality y ≤ 2x + 5?

Correct Answer: Solid

Question 10:

What does 'm' stand for in slope-intercept form?

Correct Answer: slope

Fill in the Blank Questions

Question 1:

The form y = mx + b is called ________ form.

Correct Answer: slope-intercept

Question 2:

A ______ line is used when the inequality does not include 'equal to'.

Correct Answer: dashed

Question 3:

The solution to a system of linear inequalities is the area of ______.

Correct Answer: overlap

Question 4:

When graphing y < x - 2, you would shade ______ the line.

Correct Answer: below

Question 5:

The 'b' in slope-intercept form represents the ______.

Correct Answer: y-intercept

Question 6:

Dividing both sides of an inequality by a negative number requires you to ______ the inequality sign.

Correct Answer: flip

Question 7:

If there is no overlapping shaded region, the system has ______ solution.

Correct Answer: no

Question 8:

The graph of y = 3 is a ______ line.

Correct Answer: horizontal

Question 9:

The inequality x > 4 would be represented by shading to the ______ of the vertical line.

Correct Answer: right

Question 10:

A system of inequalities requires at least ______ inequalities.

Correct Answer: two

Educational Standards

A2.A.1.9:Solve systems of linear inequalities in two variables, with a maximum of three inequalities; graph and interpret the solutions on a coordinate plane. Graphing calculators or other appropriate technology may be used. A2.A.1:Represent and solve mathematical and real-world problems using nonlinear equations, systems of linear equations, and systems of linear inequalities; interpret the solutions in the original context. A2.F.1.1:Use algebraic, interval, and set notations to specify the domain and range of various types of functions, and evaluate a function at a given point in its domain.

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