Systems of Inequalities

Chapter 1: Introduction

• Overview of Inequalities and Functions

  • Focus on systems of inequalities (two or more inequalities in variables).

  • Solutions represented graphically by overlapping regions of inequalities.

• Example Inequalities

  • First: y ≥ x² - 2x - 3

  • Second: y < 3x + 4

• Finding Solutions

  • For first inequality:

    • Set y = 0 to find x-intercepts.

    • Factor quadratic: x² - 2x - 3 = (x - 3)(x + 1) leading to x-intercepts at (3,0) and (-1,0).

  • For second inequality:

    • Set y = 0, solve: 0 = 3x + 4 ⇒ x = -4/3, x-intercept at (-4/3, 0).

  • Find y-intercepts by substituting x = 0:

    • For quadratic: y = -3 (y-intercept at (0, -3)).

    • For linear: y = 4 (y-intercept at (0, 4)).

Chapter 2: The Quadratic Function

• Vertex of Quadratic

  • Formula: x = -b/(2a); identify values: b = -2, a = 1.

  • Calculate vertex: x = 2/2 = 1.

  • Plug x back in: y = 1 - 2 - 3 = -4.

  • Vertex at (1, -4); opens upward because coefficient of x² is positive.

  • Solid line graph since the inequality includes '='.

• Characteristics of Linear Inequality

  • Slope = 3 (rise/run).

  • Graphed as a dashed line since the inequality does not include '='.

Chapter 3: Graphing Inequalities

• Graphing Quadratic Inequality

  • Key points: x-intercepts (-1, 0) and (3, 0), y-intercept (0, -3), vertex (1, -4).

  • Solid line graph.

• Graphing Linear Inequality

  • y-intercept at (0, 4), slope of 3.

  • Dashed line graph because the inequality does not include '='.

  • Test point (0, 0) for both inequalities:

    • Quadratic: 0 ≥ -3 (true), shade inside parabola.

    • Linear: 0 < 4 (true), shade below the dashed line.

Chapter 4: Representing Solutions

• Overlapping Region

  • Represents solutions for both inequalities as the area within the shaded regions.

• Alternative Graphing Tools

  • Use desmos.com for accurate graphing instead of manual plotting.

  • Example with different inequalities: y < -x² + x, y > x² - 4.

  • Finding key features for both:

    • First function: x-intercepts (0, 1); y-intercept (0, 0).

    • Second function (factors as (x-2)(x+2)): x-intercepts (2, -2); y-intercept (0, -4).

Chapter 5: Finding the Vertex

• Vertex Calculation for Both Functions

  • First Function: b = 1, a = -1 ⇒ vertex at (0.5, -0.25).

  • Second Function: vertex at (0, -4) corresponding to y-intercept.

• Graph characteristics:

  • First opens downward due to negative coefficient.

  • Both graphs are dashed lines since they do not include '='.

Chapter 6: Shading Regions

• Shading Techniques

  • Choose appropriate test points that are easy to compute.

  • First function test: zero not usable, so test (0, 1) ⇒ y < -x² + x (shading below parabola).

  • Second function test (0, 0) ⇒ y > x² - 4 (shade inside parabola).

• Overlap areas reflect the solution region for the system of inequalities.

Chapter 7: Identifying Solutions

• Testing Points for Solutions

  • Explore points inside overlapping region: (1, 1), (2, -3), (3, -6).

  • Points on dashed lines are not solutions.

  • Valid solution identified at (0, -1) as it lies within the shaded area.

Chapter 8: Conclusion

• Summary of Concepts

  • Understanding and graphing systems of inequalities is crucial for solution representation.

  • Tools are available for accurate graphing.