1. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §4.1 Solve InEqualities

2. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §2.5 → Point-Slope Line Equation  Any QUESTIONS About HomeWork §2.5 → HW-7 2.5 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 3 Bruce Mayer, PE Chabot College Mathematics Solving InEqualities  An inequality is any sentence containing  Some Examples  ANY value for a variable that makes an inequality true is called a solution. The set of all solutions is called the solution set. When all solutions of an inequality are found, we say that we have solved the inequality.

4. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 4 Bruce Mayer, PE Chabot College Mathematics Linear InEqualities  A linear inequality in one variable is an inequality that is equivalent to one of the forms that are similar to mx + b  where a and b represent real numbers and a ≠ 0.

5. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 5 Bruce Mayer, PE Chabot College Mathematics Example  Chk InEqual Soln  Determine whether 5 is a solution to a) 3x + 2 >7b) 7x − 31 ≠ 4  SOLUTION a)Substitute 5 for x to get 3(5) + 2 > 7, or 17 >7, a true statement. Thus, 5 is a solution to InEquality-a b)Substitute to get 7(5) − 31 ≠ 4, or 4≠ 4, a false statement. Thus, 5 is not a solution to InEquality-b

6. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 6 Bruce Mayer, PE Chabot College Mathematics “Dot” Graphs of InEqualities  Because solutions of inequalities like x < 4 are too numerous to list, it is helpful to make a drawing that represents all the solutions  The graph of an inequality is such a drawing. Graphs of inequalities in one variable can be drawn on a number line by shading all the points that are solutions. Open dots are used to indicate endpoints that are not solutions and Closed dots are used to indicated endpoints that are solutions

7. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 7 Bruce Mayer, PE Chabot College Mathematics Example  Graph InEqualities  Graph InEqualities: a) x < 3, b) y ≥ −4; c) −3< x ≤ 5  Soln-a) The solutions of x < 3 are those numbers less than 3. Shade all points to the left of 3 The open dot at 3 and the shading to the left indicate that 3 is NOT part of the graph, but numbers such as 1 and −2 are

8. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 8 Bruce Mayer, PE Chabot College Mathematics Example  Graph InEqualities  Graph Inequalities: a) x < 3, b) y ≥ −4; c) −3< x ≤ 5  Soln-b) The solutions of y ≥ −4 are shown on the number line by shading the point for –4 and all points to the right of −4. The closed dot at −4 indicates that −4 IS part of the graph

9. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 9 Bruce Mayer, PE Chabot College Mathematics Example  Graph InEqualities  Graph InEqualities: a) x < 3, b) y ≥ −4; c) −3< x ≤ 5  Soln-c) The inequality −3 < x ≤ 5 is read “−3 is less than x, AND x is less than or equal to 5.” Note the –OPEN dot at −3 → due to −3< x – CLOSED dot at 5 → due to x≤5

10. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 10 Bruce Mayer, PE Chabot College Mathematics Interval Notation  Interval Notation for Inequalities on Number lines can used in Place of “Dot Notation: Open Dot, ס → Left or Right, Single Parenthesis Closed Dot, ● → Left or Right, Single Square-Bracket

11. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 11 Bruce Mayer, PE Chabot College Mathematics Interval vs Dot Notation  Graph x ≥ 5 [ Dot Graph Interval Graph  Graph x < 2 ) Dot Graph Interval Graph

12. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 12 Bruce Mayer, PE Chabot College Mathematics Interval Graphing of InEqualities  If the symbol is ≤ or ≥, draw a bracket on the number line at the indicated number. If the symbol is, draw a parenthesis on the number line at the indicated number.  If the variable is greater than the indicated number, shade to the right of the indicated number. If the variable is less than the indicated number, shade to the left of the indicated number.

13. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 13 Bruce Mayer, PE Chabot College Mathematics Set Builder Notation  In MTH55 the INTERVAL form is preferred for Graphing InEqualities  A more compact alternative to InEquality Solution Graphing is SET BUILDER notation: Read as: “x such that x is… SET BUILDER Notation

14. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 14 Bruce Mayer, PE Chabot College Mathematics Compact Interval Notation  Graphed Interval Notation can be written in Compact, ShortHand form by transferring the Parenthesis or Bracket from the Graph to Enclose the InEquality.  Examples x  13 → (− , 13 ] −11< x  13 → (−11, 13] −11< x → (−11,  )

15. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 15 Bruce Mayer, PE Chabot College Mathematics Example  SetBuilder & Interval  Write the solution set in set-builder notation and interval notation, then graph the solution set. a) x ≤ −2b) n > 3  SOLUTION a) Set-builder notation: {x|x ≤ −2} Interval notation: (− , −2] Graph ]

16. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 16 Bruce Mayer, PE Chabot College Mathematics Example  Set Builder  Write the solution set in set-builder notation and interval notation, then graph the solution set. a) x ≤ −2b) n > 3  SOLUTION b) Set-builder notation: {n|n > 3} Interval notation: (3,  ) Graph (

17. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 17 Bruce Mayer, PE Chabot College Mathematics Intervals on the Real No. Line

18. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 18 Bruce Mayer, PE Chabot College Mathematics Addition Principle for InEqs  For any real numbers a, b, and c: a < b is equivalent to a + c < b + c; a ≤ b is equivalent to a + c ≤ b + c; a > b is equivalent to a + c > b + c; a ≥ b is equivalent to a + c ≥ b + c;

19. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 19 Bruce Mayer, PE Chabot College Mathematics Example  Addition Principle  Solve & Graph  Solve (get x by itself) Addition Principle Simplify to Show Solution Any number greater than −4 makes the statement true.  Graph (

20. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 20 Bruce Mayer, PE Chabot College Mathematics Multiplication Principle for InEqs  For any real numbers a and b, and for any POSITIVE number c: a < b is equivalent to ac < bc, and a > b is equivalent to ac > bc  For any real numbers a and b, and for any NEGATIVE number c: a bc, and a > b is equivalent to ac < bc  Similar statements hold for ≤ and ≥

21. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 21 Bruce Mayer, PE Chabot College Mathematics Mult. Principle Summarized  Multiplying both Sides of an Inequality by a NEGATIVE Number REVERSES the DIRECTION of the Inequality Examples

22. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 22 Bruce Mayer, PE Chabot College Mathematics Example  Solve & Graph  Solve & Graph a)b)  Soln-a) Divide Both Sides by −4  Graph Reverse Inequality as the Eqn-Divisor is NEGATIVE  The Solution Set: {y|y > −5} (

23. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 23 Bruce Mayer, PE Chabot College Mathematics Example  Solve & Graph  Soln-b) Multiply Both Sides by 7  Graph Simplify  The Solution Set: {x|x ≤ 28 } ]

24. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 24 Bruce Mayer, PE Chabot College Mathematics Example  Add & Mult Principles  Solve & Graph  SOLUTION Add ONE to Both sides Simplify Subtract x from Both Sides Divide Both Sides by 3 Simplify & Show Solution ]

25. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 25 Bruce Mayer, PE Chabot College Mathematics Example  Solve 3x − 3 > x + 7  Soln Add 3 to Both Sides Simplify  The Solution Set: {x|x > 5} Subtract x from Both Sides Divide Both Sides by 2 Simplify  Graph 8-11357-20482-268-11357-20482-268-11357-20482-26 (

26. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 26 Bruce Mayer, PE Chabot College Mathematics Example  Solve 15.4 − 3.2x < −6.76  Soln To Clear Decimals Dist. in the 100  The Solution Set: {x|x > 6.925} Simplify Subtract 1540 Simplify; Mult. By −1/320 Simplify; note that Inequality REVERSED by Neg. Mult.

27. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 27 Bruce Mayer, PE Chabot College Mathematics Example  Solve & Graph  Solve  Soln Use Distributive Law to Clear Parentheses Simplify Add 3 to Both Sides Simplify Add 2x to Both Sides Simplify; Divide Both Sides by 6

28. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 28 Bruce Mayer, PE Chabot College Mathematics Example  Solve & Graph  Solve  Soln From Last Slide  Graph  The Solution Set: {x|x ≤ –2}. Put x on R.H.S.; Note Reversed Inequality 2-7-5-3-11-8-6-22-4-802-7-5-3-11-8-6-22-4-802-7-5-3-11-8-6-22-4-80 ]

29. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 29 Bruce Mayer, PE Chabot College Mathematics Equation ↔ Inequality Equation Replace = by Inequality x = 5<x < 5 3x + 2 = 14≤3x + 2 ≤ 14 5x + 7 = 3x + 23>5x + 7 > 3x + 23 x 2 = 0≥x 2 ≥ 0

30. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 30 Bruce Mayer, PE Chabot College Mathematics Terms of the (InEquality) Trade  An inequality is a statement that one algebraic expression is less than, or is less than or equal to, another algebraic expression  The domain of a variable in an inequality is the set of ALL real numbers for which BOTH SIDES of the inequality are DEFINED.  The solutions of the inequality are the real numbers that result in a true statement when those numbers are substituted for the variable in the inequality.

31. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 31 Bruce Mayer, PE Chabot College Mathematics Terms of the (InEquality) Trade  To solve an inequality means to find all solutions of the inequality – that is, the solution set. The solution sets are intervals, and we frequently graph the solutions sets for inequalities in one variable on a number line The graph of the inequality x < 5 is the interval (− , 5) and is shown here x < 5, or (–∞, 5) ) 5

32. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 32 Bruce Mayer, PE Chabot College Mathematics Terms of the (InEquality) Trade  A conditional inequality such as x < 5 has in its domain at least one solution and at least one number that is not a solution  An inconsistent inequality is one in which no real number satisfies it.  An identity is an inequality that is satisfied by every real number in the domain.

33. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 33 Bruce Mayer, PE Chabot College Mathematics THE NON-NEGATIVE IDENTITY for ANY real number x  Because x 2 = xx is the product of either (1) two positive factors, (2) two negative factors, or (3) two zero factors, x 2 is always either a positive number or zero. That is, x 2 is never negative, or is always nonnegative

34. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 34 Bruce Mayer, PE Chabot College Mathematics Solving Linear InEqualities 1.Simplify both sides of the inequality as needed. a.Distribute to clear parentheses. b.Clear fractions or decimals by multiplying through by the LCD just as was done for equations. (This step is optional.) c.Combine like terms. 2.Use the addition principle so that all variable terms are on one side of the inequality and all constants are on the other side. Then combine like terms. 3.Use the multiplication principle to clear any remaining coefficient. If you multiply (or divide) both sides by a negative number, then reverse the direction of the inequality symbol.

35. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 35 Bruce Mayer, PE Chabot College Mathematics Example  Solve InEquality  Solve 8x + 13 > 3x − 12  SOLUTION 8x − 3x + 13 > 3x − 3x − 12 Subtract 3x from both sides. Subtract 13 from both sides. Divide both sides by 5 to isolate x. 5x + 13 > 0 – 12 5x + 13 –13 > –12 – 13 5x > −25 x > −5

36. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 36 Bruce Mayer, PE Chabot College Mathematics Example  Solve InEquality  Solve 8x + 13 > 3x – 12  SOLUTION Graph for x > −5  SOLUTION SetBuilder Notation {x|x > −5}  SOLUTION Interval Notation (−5,  ) (

37. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 37 Bruce Mayer, PE Chabot College Mathematics Example  AirCraft E.T.A.  An AirCraft is 150 miles along its path from Miami to Bermuda, cruising at 300 miles per hour, when it notifies the tower that The Twin-Turbo-Prop is now set on automatic pilot.  The entire trip is 1035 miles, and we want to determine how much time we should let pass before we become concerned that the plane has encountered Bermuda-Triangle trouble

38. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 38 Bruce Mayer, PE Chabot College Mathematics Example  AirCraft E.T.A.  Familiarize Recall the Speed Eqn: Distance = [Speed]·[time] So LET t ≡ time elapsed since plane on autopilot  Translate 300t = distance plane flown in t hours on AutoPilot 150 + 300t = plane’s distance from Miami after t hours on AutoPilot

39. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 39 Bruce Mayer, PE Chabot College Mathematics Example  AirCraft E.T.A.  Translate the InEquality for Worry Plane’s distance from Miami Distance from Miami to Bermuda ≥

40. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 40 Bruce Mayer, PE Chabot College Mathematics Example  AirCraft E.T.A.  Carry Out  State: Since 2.95 is roughly 3 hours, the tower will suspect trouble if the plane has not arrived in about 3 hours

41. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 41 Bruce Mayer, PE Chabot College Mathematics Example  CellPhone $Budget  You have just purchased a new cell phone. According to the terms of your agreement, you pay a flat fee of $6 per month, plus 4 cents per minute for calls.  If you want your total bill to be no more than $10 for the month, how many minutes can you use?

42. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 42 Bruce Mayer, PE Chabot College Mathematics Example  CellPhone $Budget  Familiarize: Say we use the phone 35 min per month. Then the Expense  Now that we understand the calculation LET x ≡ CellPhone usage in minutes per month

43. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 43 Bruce Mayer, PE Chabot College Mathematics Example  CellPhone $Budget  Translate:  Or, With 0.04 = 4/100

44. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 44 Bruce Mayer, PE Chabot College Mathematics Example  CellPhone $Budget  Carry Out

45. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 45 Bruce Mayer, PE Chabot College Mathematics Example  CellPhone $Budget  Check: If the phone is used for 100 minutes, you will have a total bill of $6 + $0.04(100) or $10   State: If you use no more than 100 minutes of cell phone time, your bill will be less than or equal to $10.

46. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 46 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §4.1 Exercise Set 62 (ppt), 53, 72, 80  Working Thru a Linear InEquality

47. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 47 Bruce Mayer, PE Chabot College Mathematics P4.1-62  Write InEquality for Passion greater-than, or equal-to Intimacy  Find Crossing Point  Thus Ans

48. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 48 Bruce Mayer, PE Chabot College Mathematics All Done for Today Eric Heiden Won Five Gold Medals and Set Five Olympic Records at the 1980 Winter Olympics

49. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 49 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –