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

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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 2

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

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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 4

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

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

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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 7

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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 8

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

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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) 10

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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 11

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

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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) 13

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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 14

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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 15

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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 16

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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 17

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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 18

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

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

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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 21 2

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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 22 −8

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1.5 Part 1 Solving Inequalities Solve. 3. 3(x + 3) ≥ 4(2 + x) 3x + 9 ≥ 8 + 4x – 3x – 3x 9 ≥ 8 + x 1 ≥ xwhich 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!!!! 23 1

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

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

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

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

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

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

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

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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 31

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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 32

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

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

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

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

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

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

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1-5 Part 3 Solving Inequalities Now try these problems on your own! Solve and graph. 10. 5x ≥ −15 and 2x < 4 11. −2x > 10 or x + 6 ≥ 7 Assign: p.38-40 #29-43 odd, 47 39

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1-5 Part 4 Solving Inequalities 12. 1 < 2x + 3 < 9 − 3 − 3 − 3 − 2 < 2x < 6 2 2 2 − 1 < x < 3 40 − 13

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

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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. 42

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1.6 Absolute Value Equations To Solve Absolute Value Equations: 1. Get the absolute value on a side by itself. 2. Set the expression inside the absolute bars equal to its value (the number on the other side). 3. Set the opposite of the expression inside the absolute bars equal to its value (the number on the other side). 4. Solve and check. 43

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

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

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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… 46

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

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

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

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

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1-6 Part 2 (Abs. Value) 5.|x| < 5 51 − 5 5 − 5 < x x < 5 AND x > − 5 x < 5 AND ‒ x < 5

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1-6 Part 2 (Abs. Value) 6. |x| > 5 52 x > 5OR − x > 5 x < − 5 − 5 5 x > 5OR x < − 5

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1-6 Part 2 (Abs. Value) 2x + 6 > 10 2x > 4 x > 2 −(2x + 6) > 10 −2x − 6 > 10 −2x > 16 x < −8 53 7. 2│2x + 6 │+ 6 ≥ 26 2│2x + 6 │ ≥ 20 │2x + 6 │ ≥ 10 OR x 2 2−8

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1-6 Part 2 (Abs. Value) 4x + 3 < 5 4x < 2 x < ½ − (4x + 3) < 5 − 4x − 3 < 5 − 4x < 8 x > − 2 54 8. Solve and graph. │4x + 3 │< 5 AND − 2 < x x < ½ −2 1/2

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1-6 Part 2 (Abs. Value) Assignment: pgs.46 #23, 25 – 36 all 55

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