1. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §4.4 2-Var InEqualities

2. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §4.3b → Absolute Value InEqualities  Any QUESTIONS About HomeWork §4.3b → HW-13 4.3 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 3 Bruce Mayer, PE Chabot College Mathematics Graphing InEqualities inequalityhalf-plane boundary  The graph of a linear equation is a straight line. The graph of a linear inequality is a half-plane, with a boundary that is a straight line.  To find the equation of the boundary line, we simply replace the inequality sign with an equals sign.

4. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 4 Bruce Mayer, PE Chabot College Mathematics Example  Graph y ≥ x  SOLUTION  First graph the boundary y = x. Since the inequality is greater than or equal to, the line is drawn solid and is part of the graph of the Solution x y -5 -4 -3 -2 -1 1 2 3 4 5 -3 2 -2 3 1 6 5 4 y = x -4 -5

5. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 5 Bruce Mayer, PE Chabot College Mathematics Example  Graph y ≥ x Note that in the graph each ordered pair on the half-plane above y = x contains a y-coordinate that is greater than the x-coordinate. It turns out that any point on the same side as (–2, 2) is also a solution. Thus, if one point in a half- plane is a solution, then all points in that half-plane are solutions. y x -5 -4 -3 -2 -1 1 2 3 4 5 -3 2 -2 3 1 6 5 4 y = x -4 -5

6. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 6 Bruce Mayer, PE Chabot College Mathematics Example  Graph y ≥ x Finish drawing the solution set by shading the half-plane above y = x. x y -5 -4 -3 -2 -1 1 2 3 4 5 -3 2 -2 3 1 6 5 4 y = x -4 -5 For any point here, y > x. For any point here, y = x. The complete solution set consists of the shaded half-plane as well as the boundary itself which is drawn solid

7. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 7 Bruce Mayer, PE Chabot College Mathematics Example  Graph y < 3 − 8x  SOLUTION  Since the inequality sign is <, points on the line y = 3 – 8x do not represent solutions of the inequality, so the line is dashed. x y -5 -4 -3 -2 -1 1 2 3 4 5 -3 2 -2 3 1 6 5 4 y = 3 – 8x -4 -5 (3, 1)  Using (3, 1) as a test point, we see that it is NOT a solution:  Thus points in the other ½-plane are solns

8. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 8 Bruce Mayer, PE Chabot College Mathematics Graphing Linear InEqualities 1.Replace the inequality sign with an equals sign and graph this line as the boundary. If the inequality symbol is, draw the line dashed. If the inequality symbol is ≥ or ≤, draw the line solid. 2.The graph of the inequality consists of a half-plane on one side of the line and, if the line is solid, the line is part of the Solution as well

9. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 9 Bruce Mayer, PE Chabot College Mathematics Graphing Linear InEqualities 3.Shade Above or Below the Line If the inequality is of the form y < mx + b or y ≤ mx + b shade below the line. If the inequality is of the form y > mx + b or y ≥ mx + b shade above the line. 4.If y is not isolated, either solve for y and graph as in step-3 or simply graph the boundary and use a test point. If the test point is a solution, shade the half-plane containing the point. If it is not a solution, shade the other half-plane

10. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 10 Bruce Mayer, PE Chabot College Mathematics Example  Graph  Draw Graph and test (3,3) = (x test, y test ) x y -5 -4 -3 -2 -1 1 2 3 4 5 -3 2 -2 3 1 6 5 4 y = (1/6)x – 1 -4 -5 (3,3)  Check Location of Test Value y test > (1/6)·x test − 1 ¿? 3 > (1/6)(3) − 1 ¿? 3 > 2 − 1   Since 3 > 1 the pt (3,3) IS a Soln, so shade on that side

11. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 11 Bruce Mayer, PE Chabot College Mathematics Example  Graph x ≥ −3  Draw Graph x y -5 -4 -3 -2 -1 1 2 3 4 5 -3 2 -2 3 1 6 5 4 -4 -5  Test (4,−2) & (1, 3) (4,−2) (1,3)  Since both 4 & 1 are greater than −3, then points to the right of the line are solutions

12. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 12 Bruce Mayer, PE Chabot College Mathematics Systems of Linear Equations  To graph a system of equations, we graph the individual equations and then find the intersection of the individual graphs. We do the same thing for a system of inequalities, that is, we graph each inequality and find the intersection of the individual Half-Plane graphs.

13. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 13 Bruce Mayer, PE Chabot College Mathematics Example  x + y > 3 & x − y ≤ 3  SOLUTION  First graph x + y > 3 in red. x y -5 -4 -3 -2 -1 1 2 3 4 5 -3 2 -2 3 1 6 5 4 -4 -5 y > −x + 3

14. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 14 Bruce Mayer, PE Chabot College Mathematics Example  x + y > 3 & x − y ≤ 3  SOLUTION  Next graph x − y ≤ 3 in blue x y -5 -4 -3 -2 -1 1 2 3 4 5 -3 2 -2 3 1 6 5 4 -4 -5 y ≥ x − 3

15. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 15 Bruce Mayer, PE Chabot College Mathematics Example  x + y > 3 & x − y ≤ 3  SOLUTION  Now find the intersection of the regions x y -5 -4 -3 -2 -1 1 2 3 4 5 -3 2 -2 3 1 6 5 4 -4 -5  The Solution is the OverLapping Region CLOSED dot indicates that the Intersection is Part of the Soln Solution set to the system

16. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 16 Bruce Mayer, PE Chabot College Mathematics Example  Graph −1 < y < 5  SOLUTION  Break into Two Inequalities and Graph −1 < y y < 5  The Solution is the OverLapping Region x y -5 -4 -3 -2 -1 1 2 3 4 5 -3 2 -2 3 1 6 5 4 -4 -5 Solution set

17. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 17 Bruce Mayer, PE Chabot College Mathematics Intersection of Two Inequalities  Graph 3x + 4y ≥ 12 and y > 2  Graph Each InEquality Separately

18. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 18 Bruce Mayer, PE Chabot College Mathematics Intersection of Two Inequalities  Graph 3x+4y≥12 and y>2  Shade Region(s) common to BOTH

19. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 19 Bruce Mayer, PE Chabot College Mathematics Union of Two Inequalities  Graph 3x + 4y ≥ 12 or y > 2  Again Graph Each InEquality Separately

20. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 20 Bruce Mayer, PE Chabot College Mathematics Union of Two Inequalities  Graph 3x+4y≥12 or y>2  Shade Region(s) covered by EITHER soln

21. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 21 Bruce Mayer, PE Chabot College Mathematics Graphing a System of InEquals  A system of inequalities may have a graph that consists of a polygon and its interior.  To construct the PolyGon we find the CoOrdinates for the corners, or vertices (singular vertex), of such a graph

22. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 22 Bruce Mayer, PE Chabot College Mathematics Example  Graph of System  Graph System Red Blue Green x y -5 -4 -3 -2 -1 1 2 3 4 5 -3 2 -2 3 1 6 5 4 -4 -5 (3, 5) (3, –3) (–1, 1 )  Draw Graph 3 Lines Intersecting at 3 locations

23. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 23 Bruce Mayer, PE Chabot College Mathematics Example  Graph of System  Graph System Red Blue Green x y -5 -4 -3 -2 -1 1 2 3 4 5 -3 2 -2 3 1 6 5 4 -4 -5  The Solution is the Enclosed Region; a PolyGon A TriAngle in this case –Check that, say, (2, 2) works in all three of the InEqualities

24. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 24 Bruce Mayer, PE Chabot College Mathematics Example  Find Vertices  Graph the following system of inequalities and find the coordinates of any vertices formed:  Graph the related equations using solid lines.  Shade the region common to all three solution sets.

25. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 25 Bruce Mayer, PE Chabot College Mathematics Example  Find Vertices  To find the vertices, we solve three systems of 2-equations.  The system of equations from inequalities (1) and (2) y + 2 = 0& −x + y = 2  Solving find Vertex pt (−4, −2)  The system of equations from inequalities (1) and (3): y + 2 = 0 &x + y = 0

26. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 26 Bruce Mayer, PE Chabot College Mathematics Example  Find Vertices  The Vertex for The system of equations from inequalities (1) & (3): (2, −2)  The system of equations from inequalities (2) and (3): −x + y = 2& x + y = 0  The Peak Vertex Point is (−1, 1) (−4,−2) (2,−2) (−1,−1)

27. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 27 Bruce Mayer, PE Chabot College Mathematics Example  Graph of System  Graph the following system. Find the coordinates of any vertices formed.  Graph by Lines  The CoOrd of the vertices are: (0, 3), (0, 4), (3, 4) and (3, 1)

28. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 28 Bruce Mayer, PE Chabot College Mathematics Types of Eqns & InEquals  Graph

29. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 29 Bruce Mayer, PE Chabot College Mathematics Types of Eqns & InEquals  Graph

30. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 30 Bruce Mayer, PE Chabot College Mathematics Types of Eqns & InEquals  Graph

31. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 31 Bruce Mayer, PE Chabot College Mathematics Types of Eqns & InEquals  Graph ≥

32. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 32 Bruce Mayer, PE Chabot College Mathematics Types of Eqns & InEquals  Graph

33. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 33 Bruce Mayer, PE Chabot College Mathematics Types of Eqns & InEquals  Graph ≤ ≤ ≥

34. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 34 Bruce Mayer, PE Chabot College Mathematics Example  PopCorn Revenue  A popcorn stand in an amusement park sells two sizes of popcorn. The large size sells for $4.00 and the smaller for $3.00 The park management feels that the stand needs to have a total revenue from popcorn sales of at least $400 each day to be profitable a)Write an inequality that describes the amount of revenue the stand must make to be profitable. b) Graph the inequality. c)Find two combinations of large and small popcorns that must be sold to be profitable

35. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 35 Bruce Mayer, PE Chabot College Mathematics Example  PopCorn Revenue  Translate by Tabulation CategoryPriceNumber SoldRevenue Large4.00x4x4x Small3.00y3y3y a)The total revenue would be found by the expression 4x + 3y. If that total revenue must be at least $400, then we can write the following inequality: 4x + 3y ≥ 400

36. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 36 Bruce Mayer, PE Chabot College Mathematics Example  PopCorn Revenue b)Graph 4x + 3y ≥ 400

37. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 37 Bruce Mayer, PE Chabot College Mathematics Example  PopCorn Revenue c)We assume that fractions of a particular size are not sold, so we will only consider whole number combinations. One combination is 100 large and 0 small popcorns which is exactly $400. A second combination is 130 large and 40 small, which gives a total revenue of $640.

38. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 38 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §4.4 Exercise Set 46 (ppt), 62  PopCorn Bag & Bucket Sizes

39. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 39 Bruce Mayer, PE Chabot College Mathematics P4.4-46 Graph System  Graph 2x + y ≤ 6  Test (0,0) 2(0)+0 ≤ 6? 0 ≤ 6   Shade BELOW Line

40. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 40 Bruce Mayer, PE Chabot College Mathematics P4.4-46 Graph System  Graph x + y ≥ 2  Test (0,0) 0+0 ≥ 2? 0 ≥ 2   Shade ABOVE Line

41. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 41 Bruce Mayer, PE Chabot College Mathematics P4.4-46 Graph System  Graph 1 ≤ x ≤ 2  Test (0,0) 1 ≤ 0 ≤ 2   Test (1.5,0) 1 ≤ 1.5 ≤ 2   Shade BETWEEN Lines

42. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 42 Bruce Mayer, PE Chabot College Mathematics P4.4-46 Graph System  Graph y ≤ 3  Test (0,0) 0 ≤ 3   Shade BELOW Line

43. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 43 Bruce Mayer, PE Chabot College Mathematics P4.4-46 Graph System  Now Check For OverLap Region  Found One; a five sided PolyGon

44. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 44 Bruce Mayer, PE Chabot College Mathematics P4.4-46 Graph System  Thus Solution

45. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 45 Bruce Mayer, PE Chabot College Mathematics All Done for Today Healthy Heart WorkOut

46. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 46 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

47. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 47 Bruce Mayer, PE Chabot College Mathematics Graph y = |x|  Make T-table

48. BMayer@ChabotCollege.edu MTH55_Lec-19_sec_4-4_2Var_InEqualities.ppt 48 Bruce Mayer, PE Chabot College Mathematics