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Understanding and Expanding Inequalities: Quadratic and Linear Concepts in Pre-Calculus

This resource covers the expansion and understanding of various types of inequalities, including one-variable quadratic inequalities and two-variable linear and quadratic inequalities. Learn the four forms of linear inequalities and how they describe regions on the Cartesian plane. The description of solution sets, boundary lines, and the concept of solution regions is thoroughly explained. Practice problems and key examples are included to enhance comprehension. Perfect for pre-calculus studies, this guide serves as a comprehensive overview of inequalities.

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Understanding and Expanding Inequalities: Quadratic and Linear Concepts in Pre-Calculus

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  1. E. Math is > the Rest Pre-Calculus 20 P20.9 Expand and demonstrate understanding of inequalities including: one-variable quadratic inequalities two-variable linear and quadratic inequalities.

  2. Key Terms:

  3. 1. Two Variable Linear Inequalities • P20.9 • Expand and demonstrate understanding of inequalities including: • one-variable quadratic inequalities • two-variable linear and quadratic inequalities.

  4. 1. Two Variable Linear Inequalities • A Linear Inequalities comes in 4 forms:

  5. A linear Inequality is describing a region on a Cartesian plane • An order pair (x,y) is a solution to the Inequality if it satisfies the equations • The set of points that satisfy the Inequality is called the Solution Region • The line related to the Inequality Ax + Bx = C called the boundary divides the Cartesian plane into to regions.

  6. For the solution region Ax + Bx ≥ C is true • For the solution region Ax + Bx ≥ C is false

  7. Example 1

  8. Example 2

  9. Example 3

  10. Example 4

  11. Key Ideas p.471

  12. Practice • Ex. 9.1 (p.472) #1-17 #8-21

  13. 2. Quadratic Inequalities in One Variable • P20.9 • Expand and demonstrate understanding of inequalities including: • one-variable quadratic inequalities • two-variable linear and quadratic inequalities.

  14. Investigate p. 476

  15. Quadratic Inequalities can be written with any inequality sign.

  16. You can solve a quadratic inequality graphically or algebraically • The solution set to a Quadratic Inequality in one variable can have 0, 1, Infinite Solutions

  17. Example 1

  18. Example 2

  19. Example 3

  20. Example 4

  21. Key Ideas p. 484

  22. Practice • Ex. 9.2 (p.484) #1-17 #9-20

  23. 3. Two Variable Quadratic Inequalities • P20.9 • Expand and demonstrate understanding of inequalities including: • one-variable quadratic inequalities • two-variable linear and quadratic inequalities.

  24. 3. Two Variable Quadratic Inequalities • You can express a quadratic inequality in two variables in one of the following forms:

  25. A Quadratic Inequality is 2 variables represents a region of the Cartesian plane with a parabola as the boundary. • The graph of Quadratic Inequality is the set of points (x,y) that are solutions to the inequalities.

  26. The boundary is set by y= x2-2x-3 and because its < , the boundary has dashed line because those points are not inclined in solutions • To determine the solution region use a test point. Lets try (0,0).

  27. Example 1

  28. Example 2

  29. Example 3

  30. Example 4

  31. Key Ideas p.496

  32. Practice • Ex. 9.3 (p.496) #1-15 #6-18

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