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Chapter 1: Expressions, Equations, & Inequalities

Chapter 1: Expressions, Equations, & Inequalities. Sections 1.3 – 1.6. 1.3 Algebraic Expressions. Algebraic Expression : contains numbers, variables, and mathematical signs (no equal sign) Equation : contains numbers, variables, mathematical signs, and an EQUAL SIGN. 1.3 Algebraic Expressions.

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Chapter 1: Expressions, Equations, & Inequalities

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  1. Chapter 1: Expressions, Equations, & Inequalities Sections 1.3 – 1.6

  2. 1.3 Algebraic Expressions • Algebraic Expression: contains numbers, variables, and mathematical signs (no equal sign) • Equation: contains numbers, variables, mathematical signs, and an EQUAL SIGN

  3. 1.3 Algebraic Expressions Write an algebraic expression 1. one less than the product of six and w 6w – 1

  4. 1.3 Algebraic Expressions 2. You are on a bicycle trip. You travel 52 miles on the first day. Since then, your average rate has been 12 miles per hour. What algebraic expression models the distance traveled? Let h be the number of hours traveled. 52 + 12h

  5. 1.3 Algebraic Expressions Evaluate the following expressions 3. 2r + 5(s+6) – 1 if r = 3, s = –9 2(3) + 5(–9+6) – 1 2(3) + 5(–3) – 1 6 + – 15 – 1 – 9 – 1 – 10

  6. 1.3 Algebraic Expressions 4. c³ - d/8 if c = ¼ , d = 1 (¼)³ – 1/8 1/64 – 1/8 1/64 – 8/64 – 7/64

  7. 1.3 Algebraic Expressions 5. Tickets to a museum are $8 for adults, $5 for children, and $6 for seniors a.) What algebraic expression models the total number of dollars collected in ticket sales? 8a + 5c + 6s

  8. 1.3 Algebraic Expressions b.) If 20 adults, 16 children, and 10 senior tickets are sold one morning, how much money is collected in all? 8(20) + 5(16) + 6(10) 160 + 80 + 60 300

  9. 1.3 Algebraic Expressions Simplify 6. 2a² + 3b² + 6b² + 5a² 7a² + 9b²

  10. 1.3 Algebraic Expressions 7. –(x + 4y) + 5(3x – y) – x – 4y + 15x – 5y 14x – 9y Assign pgs: 22 – 23, #10 – 19, 20 – 26 even, 30 – 44 even, 52 (23 problems)

  11. 1.4 Solving Equations • Reflexive: a = a • Symmetric: if a = b then b = a • Transitive: if a = b and b = c, then a = c • Addition: if a = b then a + c = b + c • Subtraction: if a = b then a - c = b – c • Multiplication: if a = b then a(c) = b(c) • Division: if a = b then a ÷ c = b ÷ c

  12. 1.4 Solving Equations Solve the following equations 1. x – 8 = -10 +8 +8 x = – 2

  13. 1.4 Solving Equations 2. – 2(y – 1) = -16 + y – 2y + 2 = – 16 + y +2y +2y 2 = – 16 + 3y +16 +16 18 = 3y 3 3 y = 6 Assign Pg. 23 – 24, #53, 55, 63 – 66 Pg. 30, #10 – 24 even (14 problems)

  14. 1.4 Solving Equations Cont’d Solve 1. 1 + 5x -6 = 6x – 5 – x 5x – 5 = 5x – 5 – 5x – 5x – 5 = – 5 which means… infinite number of solutions or all real numbers

  15. 1.4 Solving Equations Cont’d 2. –x + 2(5x – 1) = 2(3x+4) + x – x + 10x – 2 = 6x + 8 + x 9x – 2 = 7x + 8 – 7x – 7x 2x – 2 = 8 + 2 +2 2x = 10 x = 5

  16. 1.4 Solving Equations Cont’d 3. What is t in terms of A in A = 1000(1+0.05t) A = 1000 + 50t – 1000 – 1000 A – 1000 = 50t 50 50 t = A – 20 50

  17. 1.4 Solving Equations Cont’d 4. Solve A = ½ (b+ c) for b 2(A) = 2 (½)(b + c) 2A = b + c – c – c 2A - c = b b = 2A – c Assign pgs: 30–31, #28 – 36, 38, 41, 42, 46, 48, 49, 61 16 problems

  18. 1.5 Part 1 Solving Inequalities • Transitive: if a > b and b > c, then a > c • Addition: if a > b then a + c > b + c • Subtraction: if a > b then a - c > b – c • Multiplication: if a > b and c > 0 then a(c) > b(c) if a > b and c < 0 then a(c) < b(c) • Division: if a > b and c > 0 then a ÷ c > b ÷ c if a > b and c < 0 then a ÷ c < b ÷ c

  19. 1.5 Part 1 Solving Inequalities *If you multiply or divide by a negative number, FLIP THE ARROW! Graphing: mean open dots >, < mean closed dots ≥, ≤

  20. 1.5 Part 1 Solving Inequalities Graph x > 3. Graph 3 < x. Graph 4 < x. 3 3 4

  21. 1.5 Part 1 Solving Inequalities 1. Solve the inequality and graph the solution. 4(x – 7) > −20 4x – 28 > –20 +28 +28 4x > 8 4 4 x > 2 2

  22. 1.5 Part 1 Solving Inequalities 2. 4(−n – 2) – 6 >18 – 4n – 8 – 6 > 18 – 4n – 14 > 18 + 14 +14 – 4n > 32 – 4 – 4 n < – 8 −8

  23. 1.5 Part 1 Solving Inequalities Solve. 3. 3(x + 3) ≥ 4(2 + x) 3x + 9 ≥ 8 + 4x – 3x – 3x 9 ≥ 8 + x 1 ≥ x which can also be written as x ≤ 1 Assign pgs.38-40: #14-23 all, 68,69,71-78 all Reminder: QUIZ (1.3 – 1.4) TOMORROW!!!! 1

  24. 1-5 Part 2 Solving Inequalities 4. What inequality represents the sentence? • 5 fewer than the product of seven and a number is no more than 50. 7n – 5 < 50

  25. 1-5 Part 2 Solving Inequalities What inequality represents the sentence? • The quotient of a number and 6 is at least 10.

  26. 1-5 Part 2 Solving Inequalities 5. −½(y + 3) ≥ 1/3y – 4

  27. 1-5 Part 2 Solving Inequalities 5 (cont’d) y ≤ 3

  28. 1-5 Part 2 Solving Inequalities 5 (cont’d) y < 3 Assign pgs 38 – 39: # 10-13 all, 24,27,45,46 8 problems

  29. 1-5 Part 2 Solving Inequalities 5. −½(y + 3) ≥ 1/3y – 4 –3y – 9 ≥ 2y – 24 –2y –2y

  30. 1-5 Part 2 Solving Inequalities 5 (cont’d) –5y – 9 ≥ –24 +9 +9 –5y ≥ –15 y ≤ –3 Assign pgs 38 – 39: # 10-13 all, 24,27,45,46 8 problems

  31. 1-5 Part 3 Solving Inequalities Solve 6. 9 – x – 5 < -x + 4 – x + 4 < – x + 4 + x + x 4 < 4 which means… No Solution

  32. 1-5 Part 3 Solving Inequalities Solve 7. 9 – x – 5 ≤ − x + 4 – x + 4 ≤ – x + 4 + x + x 4 ≤ 4 which means… All Real Numbers

  33. 1-5 Part 3 Solving Inequalities Compound Inequality: Two inequalities joined together by the word “and” or the word “or”

  34. 1-5 Part 3 Solving Inequalities “and” The solution must be true for both inequalities at the same time. (usually shades in the middle)

  35. 1-5 Part 3 Solving Inequalities 8. ½a < 3 and–3a + 5 < 8 2(½a) < 2(3) a < 6 – 3a + 5 < 8 −5 −5 – 3a < 3 – 3 – 3 a > – 1 and a < 6

  36. 1-5 Part 3 Solving Inequalities 8. (cont’d) ½a < 3 and–3a + 5 < 8 Smallest number a < 6 a > − 1 and This is the solution!! – 1 < a a < 6 − 1 6

  37. 1-5 Part 3 Solving Inequalities “or” The solution will make any or all parts of the inequalities true. (usually shades to the outside)

  38. 1-5 Part 3 Solving Inequalities 9. ½a > 3 or–3a + 5 > 8 a > 6 or a < − 1 All of this is the solution!!! − 1 6

  39. 1-5 Part 3 Solving Inequalities Now try these problems on your own! Solve and graph. 10. 5x ≥ −15 and 2x < 4 • −2x > 10 or x + 6 ≥ 7 Assign: p.38-40 #29-43 odd, 47

  40. 1-5 Part 4 Solving Inequalities 12. 1 < 2x + 3 < 9 − 3 − 3 − 3 − 2 < 2x <6 2 2 2 −1 < x < 3 − 1 3

  41. 1-5 Part 4 Solving Inequalities Assign: p.38-40 #28-42 even, 55,59,67

  42. 1.6 Absolute Value Equations Absolute value: the distance from 0 on a number line │5 │= 5 │−5 │= 5 Notice that either a number OR its opposite have the same absolute value.

  43. 1.6 Absolute Value Equations To Solve Absolute Value Equations: • Get the absolute value on a side by itself. • Set the expression inside the absolute bars equal to its value (the number on the other side). • Set the opposite of the expression inside the absolute bars equal to its value (the number on the other side). • Solve and check.

  44. 1.6 Absolute Value Equations x = 5 − x = 5 − 1 − 1 x = −5 1. Solve. |x| = 5 SOLUTION x = 5,− 5 x = ± 5

  45. 1.6 Absolute Value Equations Solve. 2. │2x + 5 │= 9 −(2x + 5) = 9 • −2x − 5 = 9 • − 2x = 14 • x = − 7 • 2x + 5 = 9 • 2x = 4 • x = 2 x = 2, −7

  46. 1.6 Absolute Value Equations 3. ½│2x − 4 │ − 2 = 6 +2 +2 ½│2x − 4 │= 8 2 ∙ (½│2x − 4 │) = 2 ∙ (8) │2x − 4 │= 16 Continued on next slide…

  47. 1.6 Absolute Value Equations 3. continued │2x − 4 │= 16 − (2x – 4) = 16 −2x + 4 = 16 −2x = 12 x = −6 2x – 4 = 16 2x = 20 x = 10 x = 10, − 6

  48. 1.6 Absolute Value Equations 4. |3x| = −9 − 3x = −9 x = 3 3x = − 9 x = − 3 NO SOLUTION!!! WHY ???????????

  49. 1.6 Absolute Value Equations Assignment pgs.46 #10 – 18 all, 22

  50. 1-6 Part 2 (Abs. Value) Less than (and) Greater (or) an an er or

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