Math 2 Variable Manipulation Part 3 COMPLEX NUMBERS A Complex Number is a combination of a Real Number and an Imaginary Number: 1 Examples: 1 + i 39 + 3i 0.8 2.2i 2 + πi 2 + i/2 A Complex Number is just two numbers added together (a Real and an Imaginary Number). But either part can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers. Complex Number Real Part Imaginary Part 3 + 2i 3 2 5 5 0 6i 0 6 Add & Subtract Complex Numbers To add two complex numbers, add each part separately (remember: numbers must be the same type to add or subtract). (a+bi) + (c+di) = (a+c) + (b+d)i Example: (3 + 2i) + (1 + 7i) = (4 + 9i) Multiplying Complex Numbers Each part of the first complex number gets multiplied by each part of the second complex number Just use "FOIL", which stands for "Firsts, Outers, Inners, Lasts" Firsts: a c Outers: a di Inners: bi c Lasts: bi di Example: (3 + 2i)(1 + 7i) Solution: Example: (1 + i) 2 Solution: (a+bi)(c+di) = ac + adi + bci + bdi 2 (3 + 2i)(1 + 7i) = 3 1 + 3 7i + 2i 1+ 2i 7i = 3 + 21i + 2i + 14i 2 = 3 + 21i + 2i 14 (because i 2 = 1) = 11 + 23i (1 + i)2 = (1 + i)(1 + i) = 1 1 + 1 i + 1 i + i 2 = 1 + 2i - 1 (because i 2 = 1) = 0 + 2i
But There is a Quicker Way! Use this rule: (a+bi)(c+di) = (ac bd) + (ad+bc)i 2 Example: (3 + 2i)(1 + 7i) = (3 1 2 7) + (3 7 + 2 1)i = 11 + 23i Why Does That Rule Work? It is just the "FOIL" method after a little work: (a+bi)(c+di) = ac + adi + bci + bdi 2 FOIL method = ac + adi + bci bd (because i 2 = 1) = (ac bd) + (ad + bc)i (gathering like terms) And there we have the (ac bd) + (ad + bc)i pattern. This rule is certainly faster, but if you forget it, just remember the FOIL method. Example: i 2 Solution: i can also be written with a real and imaginary part as 0 + i i 2 = (0 + i)2 = (0 + i)(0 + i) = (0 0 1 1) + (0 1 + 1 0)i = 1 + 0i = 1 And that agrees nicely with the definition that i 2 = 1 Dividing Complex Numbers To take divide complex numbers, multiply both numerator and denominator by the complex conjugate. Example: What is 2+5i 3+i as a single complex number? Solution: Multiply both numerator and denominator by the complex conjugate of 3 + i (which is 3 - i) COMPLEX NUMBERS IN QUADRATIC EQUATIONS Example: What are the roots of the quadratic equation x² - 2x + 3 = 0 Solution: Use the quadratic equation formula x = b± b2 4ac 2a with a = 1, b = -2 and c = 3 ( 2)± ( 2)² 4 1 3 x = 2 1 = 2 ± 8 2 = 1 + i 2 or 1 - i 2 Your calculator can be a really handy tool for problems dealing with imaginary and irrational numbers.
Sample Questions? 1. (2 + i)(2 - i)? 2 3 2. (3 + i) 2? 3. What is (3-5i) + (-4 + 7i)? 4. What is 7 3i as a single complex number? 5 2i 5. What are the roots of the quadratic equation x² + 4x + 7 = 0? 6. In the complex numbers, where i 2 = -1, 1 x (1 i) =? (1+i) (1+i) 7. For i 2 = 1, (4 + i) 2 =? 8. What is the product of the complex numbers ( 3i + 4) and (3i + 4)? 9. For the imaginary number i, which of the following is a possible value of i n if n is an integer less than 5? A. 0 B. -1 C. -2 D. -3 E. -4
4 SOLVING AN EQUATION THAT INCLUDES ABSOLUTE VALUE SIGNS To solve an equation that includes absolute value signs, think about the two different cases-one where what is inside the absolute value sign equals a positive number, and one where it equals a negative number. For example, to solve the equation x 12 = 3, think of it as two equations: x - 12 = 3 or x - 12 = -3 x = 15 or 9 Just like with quadratic equations, equations with absolute value signs will have two possible solutions. Example: Given the equation y 2 11-2 = 0, which of the following is a solution but NOT a rational number? a. 11 13 b. 4 13 c. 2 13 d. 13 e. 0 Solution: To find the solutions, solve the equation for y. First isolate the absolute value expression y2 11 = 2. Remember that you will have to create two different equations as you remove the absolute value sign: y 2 11 = 2 and y 2 11 = -2. Solve each of these equations to find that y 2 = 13 and y 2 = 9 so y = ±3 and y = ± 13. The irrational solutions are ± 13. Example: Solve the following equation: 2x 3 4 = 3 Solution: 2x 3 = 7 Now I'll clear the absolute-value bars by splitting the equation into its two cases, one for each sign: (2x 3) = 7 or (2x 3) = 7 2x 3 = 7 or 2x + 3 = 7 2x = 10 or 2x = 4 x = 5 or x = 2 So the solution is x = 2, 5 Example: How many solutions does the equation have? -2 = q + 2 Solution: No solution because inequalities can never be negative Sample Questions: 10. The expression 2 14 - -25 is equal to: 11. Solve the following equation: 2x 1 = 5 12. If x = x + 12, then x =?
13. For real numbers a and b, when is the equation a + b = a b true? a. Always b. Only when a = b c. Only when a = 0 and b = 0 d. Only when a = 0 or b = 0 e. Never 5 14. If x > y, which of the following is the solution statement for x when y = -4? a. x is any real number. b. x > 4 c. x < 4 d. -4 < x < 4 e. x > 4 or x < -4 15. For real numbers r and s, when is the equation r s = r + s true? f. Always g. Only when r = s h. Only when r = 0 or s = 0 i. Only when r > 0 and s < 0 j. Never ABSOLUTE VALUE (LESS THAN) In general, if an inequality is in the form: a < b, then the solution will be in the form: b < a < b for any argument a. Example: a < 3. Solution: For that inequality to be true, what values could a have? Geometrically a is less than 3 units from 0. Therefore, 3 < a < 3. The inequality will be true if a has any value between 3 and 3. Example: For which values of x will this inequality be true? 2x 1 < 5. Solution: The argument, 2x 1, will fall between 5 and 5: 5 < 2x 1 < 5. We must isolate x. First, add 1 to each term of the inequality: 5 + 1 < 2x < 5 + 1 4 < 2x < 6 Now divide each term by 2: 2 < x < 3. The inequality will be true for any value of x in that interval.
Sample Questions: 16. Solve this inequality for x: x + 2 < 7 6 17. Solve this inequality for x: 3x 5 < 10. 18. Solve this inequality for x: 1 + 2x < 9. ABSOLUTE VALUE (GREATER THAN) If the inequality is in the form a > b (and b > 0), then a > b or a < b. Example: a > 3 For which values of a will this be true? Geometrically, a > 3 or a < 3. 19. Solve for x: x > 5. 20. For which values of x will this be true? x + 2 > 7. 21. Solve for x: 2x + 5 > 9. 22. If 6 2 x > 9, which of the following is a possible value of x? a. -2 b. -1 c. 0 d. 4 e. 7 23. If 5 2x >5, which of the following is a possible value of x? a. 2 b. 3 c. 4 d. 5 e. 6
7 SOLVING AN INEQUALITY Solving inequalities works just the same as solving equations--do the same thing to both sides, until the variable you are solving for is isolated--with one exception: When you multiply or divide both sides by a negative number, you must reverse the inequality sign. Example: 5x - 3 < 7 Solution: Add 3 to both sides to get 5x < 10. Now divide both sides by 5 to get x < 2. Example: -5x + 7 < -3 Solution: Subtract 7 from both sides to get -5x < -10. Now divide both sides by -5, remembering to reverse the inequality sign: x > 2. Sample Questions: 24. If 5 times a number x is subtracted from 15, the result is negative. Which of the following gives the possible value(s) for x? a. All x < 3 b. All x > 3 c. 10 only d. 3 only e. 0 only 25. Which of the following inequalities defines the solution set for the inequality 23 6x 5? a. x 3 b. x 3 c. x 6 d. x 3 e. x 6 26. Which of the following is the set of all real numbers x such that x 3 < x 5? a. The empty set b. The set containing only zero c. The set containing all nonnegative real numbers d. The set containing all negative real numbers e. The set containing all real numbers 27. For which values of x will 3(x + 4) 9(4 + x)? a. x -4 b. x -4 c. x -16 d. x 4 e. x -16 28. What is the solution set of 3a 2 7? a. {a: a 3} b. {a: - 5 3 a 3} c. {a: - 5 3 a 3} d. {a: - 5 3 a 3} e. {a: - 5 3 a 3}
29. Which of the following intervals contains the solution to the equation x -2 = 2x+5? a. -6 < x < 11 b. 11 x < 15 c. 6 < x 10 d. -5 < x -3 e. -11 x -2 3 8 30. The inequality 6(x + 2) > 7(x 5) is equivalent to which of the following inequalities? a. x < -23 b. x < 7 c. x < 17 d. x < 37 e. x < 47 31. If x 2 3 13, what is the greatest real value that x can have? a. 10 b. 5 c. 4 d. 3 e. 0 32. What are the real solutions to the equation x 2 + 2 x - 3 = 0? a. ± 1 b. ± 3 c. 1 and 3 d. -1 and -3 e. ± 1 and ± 3 OTHER VARIABLE MANIPULATION QUESTIONS WITH INEQUALITIES Many variable manipulation type problems can contain inequalities. Plug in, solve for x, translate into English, fractions, absolute value, etc. Just solve the problems as regular equations with one exception: When you multiply or divide both sides by a negative number, remember to reverse the inequality sign. Sample Questions: 33. If 3 times a number x is added to 12, the result is negative. Which of the following gives the possible value(s) for x? a. All x > 4 b. All x < -4 c. 36 only d. 4 only e. 0 only 34. What is the largest integer value of t that satisfies the inequality 24 > t 30 24?
9 35. Passes to Renaissance Faire cost $9 when purchased online and $12 when purchased in person. The group sponsoring the fair would like to make at least $4,000 from sales of passes. If 240 passes were sold online, what is the minimum number of tickets that must be sold in person in order for the group to meet its goal? 36. How many different integer values of a satisfy the inequality 1 11 < 2 a < 1 8? 37. The temperature, t, in degrees Fahrenheit, in a certain city on a certain spring day satisfies the inequality t 34 40. Which of the following temperatures, in degrees Farenheit, is NOT in this range? a. 74 b. 16 c. 0 d. -6 e. -8 38. It costs a dollars for an adult ticket to a reggae concert and s dollars for a student ticket. The difference between the cost of 12 adult tickets and 18 student tickets is $36. Which of the following equations represents this relationship between a and s? a. 12a 18s = 36 b. 216as = 36 c. 12a 18s = 36 d. 12a + 18s = 36 e. 18a 12s = 36 39. What is the smallest possible integer for which 15% of that integer is greater than 2.3? 40. If a + b = 25 and a > 4, then which of the following MUST be true? a. a = 22 b. b < 21 c. b > 4 d. b = 0 e. a < 25 41. If m > 0 and n < 0, then m n: a. Is always positive b. Is always negative c. Is always zero d. Cannot be zero, but can be any real number other than zero e. Can be any real number
10 42. Considering all values of a and b for which a + b is at most 9, a is at least 2, and b is at least -2, what is the minimum value of b a? 43. If X, Y and Z are real numbers, and XYZ = 1, then which of the following conditions must be true? a. XZ = 1/Y b. X, Y, and Z > 0 c. Either X = 1, Y = 1, or Z = 1 d. Either X = 0, Y = 0, or Z = 0 e. Either X < 1, Y < 1, or Z < 1 44. If ghjk = 24 and ghkl = 0, which of the following must be true? a. g > 0 b. h > 0 c. j = 0 d. k = 0 e. l = 0 45. If y z, what are the real values of x that make the following inequality true? xy xz 3y 3z < 0 a. All negative real numbers b. All positive real numbers c. - 1 3 only d. 1 3 only e. 3 only 46. For all values x, y, and z, if x y and y z, which of the following CANNOT be true? I. x = z II. x > z III. x < z a. I only b. II only c. III only d. I and II only e. I, II, and III 47. If x and y are any real numbers such that 0 < x < 2 < y, which of these must be true? a. x < (xy)/2 < y b. 0 < xy < 2x c. x < xy < 2 d. 0 < xy < 2 e. xy < y 48. If r and s can be any integers such that s > 10 and 2r + s = 15, which of the following is the solution set for r? a. r 3 b. r 0 c. r 2.5 d. r 0 e. r 2.5
11 49. If 0 < pr < 1, then which of the following CANNOT be true? a. P < 0 and r < 0 b. P < -1 and r < 0 c. P < -1 and r < -1 d. P < 1 and r < 1 e. P < 1 and r > 0 50. If x and y are real numbers such that x > 1 and y < -1, then which of the following inequalities must be true? a. x > 1 y b. x 2 > y c. x 3 5 > y 3 5 d. x 2 + 1 > y 2 + 1 e. x -2 > y -2 51. Which of the following expressions has a positive value for all x and y such that x > 0 and y < 0? a. y x b. x + y c. x 3 y d. x2 y x e. y 2 52. If a < b, then a b is equivalent to which of the following? a. a + b b. (a + b) c. a b d. a b e. (a b) 53. If x and y are real numbers such that x > 1 and y < -1, then which of the following inequalities must be true? a. x > 1 y b. x 2 > y c. x 3-5 > y 3-5 d. x 2 + 1 > y 2 + 1 e. x 2 > y 2 54. If x < y, then x y is equivalent to which of the following? a. x + y b. (x + y) c. x y d. x y e. (x y) 55. What are the values of a and b, if any, where - a b + 4 > 0? a. a > 0 and b -4 b. a > 0 and b 4 c. a < 0 and b -4 d. a < 0 and b -4 e. a < 0 and b -4
12 GRAPHING INEQUALITIES To graph a range of values on a number line, use a thick black line over the number line, and at the end(s) of the range, use a solid circle if the point is included or an open circle if the point is not included. The figure here shows the graph of -3 < x 5. Example: 3x + 5 > 11 Solution: Solve this inequality the same way that you solve an equality. By subtracting 5 from both sides and then dividing both sides by 3, you get the expression X > 2 This can be represented on a number line as shown below. The open circle at 2 indicates that x can include every number greater than 2, but not 2 itself or anything less than 2. If we had wanted to graph x 2, the circle would have to be filled in, indicating that our graph includes 2 as well. Remember that when you multiply or divide an inequality by a negative, the sign flips. Example: Which of the following represents the range of solutions for inequality -5x -7 < x + 5? Solution: First, simplify the inequality and then figure out which of the answer choices represents a graph of the solution set of the inequality. To simplify, isolate x on one side of the inequality. -5x - 7 < x + 5 -x -x -6x - 7 < 5 +7 +7-6x < 12 Now divide both sides by -6. Remember that when you multiply or divide an inequality by a negative, the sign flips over. 6x 6 < 12 6 x > -2 Choice (B) is correct.
Sample Questions: 56. Graph the inequality 3x 6 > 6x + 9? 13 57. Graph the solution statement for the inequality shown below. 5 < 1 3x < 10 58. Solve and graph for x: 1 2x > 9. 59. Graph the solutions for the inequality (x 1)(4 x) < 0. 60. Number a, the equation x -2a = 5. On a number line, how far apart are the 2 solutions for x? 61. The solution set of which of the following equations is the set of real numbers that are 5 units from -3? a. x + 3 = 5 b. x - 3 = 5 c. x + 5 = 3 d. x - 5 = 3 e. x + 3 = -3 FACTORS WITH VARIABLE MANIPULATION Sample Questions: 62. Assuming q is a positive integer, then the difference between 14q and 5q is always divisible by: a. 5 b. 9 c. 14 d. 19 e. 70 B
Answer Key 1. 5 2. 8 + 6i 3. -1 + 2i 41 i 41 4. or - 1 i 29 29 29 5. -2 + i 3 or -2 i 3 6. (i+1) 2 or ( i 1) 2 or ( 1 i) 2 7. 15 + 8i 8. 25 9. -1 (B) 10. -13 11. 3 and -2 12. -6 13. Only when a = 0 or b = 0 14. x > 4 15. Only when r = 0 or s = 0 16. -9 < x < 5 17. -5/3 < x < 5 18. 5 < x < 4 19. x > 5 or x < 5 20. x > 5, x < 9 21. x > 2 x < -7 22. x < -3/2 or x > 15/2 23. x < 0 or x > 5 24. All x > 3 25. x 3 26. A 27. x -4 28. B 29. B 30. E 31. 4 32. ± 1 33. All x < -4 34. 19 35. 154 36. 5 37. E 38. C 39. 16 40. B 41. A 42. -13 43. A 44. E 45. A 46. B 47. A 48. E 49. C 50. C 51. E 52. E 53. C 54. E 55. D 56. Graph of x < -5 57. Graph of 3 < x < 2 58. Graph of x < -4 or x > 5 59. Graph of x < 1 or x > 4 60. 10 61. x + 3 = 5 62. 9 14