1. Algebra 1 Unit 2: Learning Goals 2.1, 2.2 & 2.3 Graphing and Analyzing One Variable Linear Inequalities Michelle A. O’Malley League Academy of Communication Arts Greenville, South Carolina

2. Learning Goal 2.1  Write and solve one, two, and multi- step inequalities in one variable, and graph the solution set on the number line, including real-world applications.

3. Learning Goal 2.1 Standards  EA1.3: Apply algebraic methods to solve problems in real-world contexts  EA4.8: Carry out procedures to solve linear inequalities for one variable algebraically and then to graph the solution  EA5.12 Analyze given information to write a linear inequality in one variable that models a given problem situation.

4. Learning Goal 2.1 Word Wall Words  Inequality  Greater than  Less than  Equal to  Not equal to  Solution set  Infinite  Discrete  Continuous

5. Inequality Review  > means greater than  ≥ means greater than or equal to  < means less than  ≤ means less than or equal to  ≠ means is not equal to  Proper way to read the above information would be 3 < X means “x is greater than 3”

6. Solution Set  A solution set of a linear inequality in one variable is the set of all values that satisfy the inequality. (The statement is true)  Note: an inequality has an infinite number of solutions. (Infinite means never ends)

7. Discrete Vs. Continuous  A domain may be discrete or continuous Discrete data is data that can be described by whole numbers or fractional values. Example: the data has an ending point such as whole numbers, which are not continuous or repeating. Continuous data is data that relate to a complete range of values on the number line. Example: The possible sizes of applies are continuous data

8. Discrete Vs. Continuous  The graph of a solution can be continuous on a number line (a ray). For example, at what temperatures will the ice in a skating pond remain frozen?  The graph of a solution can also be discrete set of points on a number line. For example, a club must have 25 or more members.

9. Graphing Inequalities  > means greater than  < means less than When the above two inequalities are graphed, the graph will show an open dot at the endpoint. This means that the value at that position is not included in the solution set.  ≥ means greater than or equal to  ≤ means less than or equal to When the above two inequalities are graphed, the graph will show a solid dot at the endpoint. This means that the value at that position is included in the solution set.

10. Solving Inequalities  When solving inequalities you follow the same methods that you used as when solving equations; however, there is one exception. When you multiply or divide both sides of the inequality by the same negative number, the direction of the inequality reverses.

11. Solving Inequalities  This is because multiplying a number by -1 moves it to the other side of the zero on the number line ( a reflection).  For example When multiplying 2 < 5 (a true statement) by -1, it will force the inequality to change to -2 > -5 to keep the statement true.

12. Solving Inequalities  However, a negative sign in an inequality does not necessarily mean that the direction of the inequality symbol should change. For example, when solving x/6 > -3, do not change the direction of the inequality. Can you see why?

13. Real World Applications Example 1  Nick’s car averages 18 miles per gallon of gasoline. If x represents the number of gallons, how much gasoline will he need to travel at least 450 miles? 18x ≥ 450 X ≥ 25 He will have to buy 25 or more gallons.

14. Real World Applications Example 2  An amusement park charges $5 for admission and $1.25 for each ride. Katie goes to the park with $25. If x represents the number of rides, what is the maximum number of rides that she can afford? 5 + 1.25x ≤ 25 1.25x ≤ 20 X ≤ 16 Katie can ride at most 16 rides.

15. Solving Inequalities by Addition and Subtraction

16. Solving Inequalities by Multiplication and Division

17. Solving Multi-Step Inequalities

18. Learning Goal 2.2  Use Venn Diagrams to represent unions and intersections of sets.

19. Learning Goal 2.2: Standards  EA1.6: Understand how algebraic relationships can be represented to concrete models, pictorial models, and diagrams

20. Learning Goal 2.2 Word Wall Words  Universal Set  Intersection  Union  Disjoint  Intersecting  Venn Diagram

21. Learning Goal 2.2 Word Wall Words  Intersection – the set of elements that belong to each of two overlapping sets.  Union – a set that is formed by combining the members of two or more sets, as represented by the symbol U. The Union contains all members previously contained in either set.  Disjoint – Not connected  Venn Diagram – a pictorial means of representing the relationships between sets.

22.  An element belongs to the intersection of two sets A and B (written A B) if and only if the element belongs to set A and to set B.  An element belongs to the union of two sets A and B (written A U B) if and only if the element belongs to set A or to set B or both A and B. Venn Diagrams

23. Venn Diagram  A Venn Diagram shows relationships among sets of data or objects. It usually consists of a rectangle that represents the universal set. Circles are show inside the rectangle to represent subsets of the universal set.

24. Venn Diagram  Set can be related to each other in three ways as illustrated in the diagrams below.  The sets can be separate (disjoint), overlapping (intersecting), or within each other (subsets). A B U A B U

25. Venn Diagram  Sets A and B are disjoint since they have no elements in common.  For example, the universal set (U) represents the set of all polygons.  Circle A represents the set of all parallelograms.  Circle B represents the set of all trapezoids A B U r1 r2r3

26. A B U Venn Diagram  Sets A and B intersect. For example, the universal set (U) represents the set of polygons.  Circle A represents the set of all rectangles.  Circle B represents the set of all rhombuses.  The intersection of the two circles is the set of all squares, which represents polygons that are both rectangles and rhombuses. r1 A B U r2r3r4 r1

27. Venn Diagram  The set B is a subset of A since all elements in B are also elements of A.  For example, the universal set (U) represents the set of all polygons.  Circle A represents the set of all rectangles.  Circle B represents the set of all squares. A B U r2 r3 r1

28.  In the figure below, Region A B is represented by (the empty set).  A U B is represented by all the elements in region 2 combined with all the elements in the region 3. Venn Diagram A B U r1 r2r3

29. A B U Venn Diagram  In the figure below, region A B is represented by elements in region 4.  A U B is represented by all the elements in region 2 combined with all the elements in regions 3 and 4. r1 A B U r2r3r4 r1

30. Venn Diagram  In the figure below, region A B is represented by region 3. A U B is represented by all the elements in region 2, which includes the elements in region 3. A B U r2 r3 r1

31. Venn Diagram  In the figure below, region 6 (r6) represents the intersection of all three sets.  For example, if the three circles represent people who like chocolate, vanilla, and strawberry ice cream, region 6 represents those people who like all three flavors. r6 CS V r2 r1 r3 r4 r5r7 r8

32. Using Venn Diagrams to Solve Problems

33. Learning Goal 2.3  Write and solve compound inequalities and graph the solution set on the number line including real- world applications.

34. Learning Goal 2.3: Standards  EA-1.5 Demonstrate an understanding of algebraic relationships by using a variety of representations (including verbal, graphic, numerical, and symbolic).  EA-4.8 Carry out procedures to solve linear inequalities for one variable algebraically and then to graph the solution. (Extension)

35. Learning Goal 2.3 Word Wall Words  Compound Inequality  Intersection  Union

36. Learning Goal 2.3  A compound inequality is two or more inequalities joined with the world “and” or with the word “or.”  To solve a compound inequality with the word “and” (a conjunction), solve each simple inequality, then find the solution that make both simple inequalities true.  This solution is the intersection of the solution sets of the simple inequalities.

37. Learning Goal 2.3  3 < X < 5 is equivalent to 3 < x and x < 5, then it is also true that 3 < 5.  It is considered good form to write the smallest numbers on the left as shown above.  Compound conjunctions can be solve by either working with the combined form or by separating it into two simple inequalities and finding the intersection of the two solutions.

38. Learning Goal 2.3  To solve a compound inequality with the word “or” (a disjunction), solve each simple inequality, then find the solutions that make either simple inequality true.  The solution is the union of the solution sets of the simple inequalities.

39. Learning Goal 2.3  “AND” means Intersection and also conjunction This means that the solution must be found in both inequalities; therefore the solution is the common area of your graphs.  “OR” means union and Disjunction This means that the solution will be found in one or the other inequalities; therefore the solution will be ALL areas graphed combined for both inequalities.

40. Learning Goal 2.3 Solve and Graph a Union Step 1: Solve each individual inequality Step 2: Graph each individual inequality solution Step 3: for a Union your compound inequality solution will be all sections combined

41. Learning Goal 2.3 Solve and Graph an Intersection Step 1: solve each individual inequality Step 2: Graph each individual inequality Step 3: The solution for the intersection will be the area that each graph has in common

42. Work Cited  Carter, John A., et. al. Glencoe Mathematics Algebra I. New York: Glencoe/McGraw-Hill, 2003.  Greenville County Schools Math Curriculum Guide  Gizmos http://www.explorelearning.com http://www.explorelearning.com  Math Use’s Handbook: Hot Words Hot Topics. New York: Glencoe McGraw- Hill, 1998.