CK-12 Algebra II with Trigonometry Concepts 1

Transcription

1 1.1 Subsets of Real Numbers 1. Rational Number. Irrational Number. Rational Number 4. Whole Number 5. Integer 6. Irrational Number 7. Real, Rational, Integer, Whole, and Natural Number 8. Real and Rational Number 9. Real and Irrational Number 10. True 11. True 1. False 1. True 14. True 15. False CK-1 Algebra II with Trigonometry Concepts 1

2 1. Ordering Real Numbers 1. ans ans ans ans , 4,,, ,, 0,, , 4,,.68, ,,, 5, > 10. < 11. > 1. = 1. = 14. < 15. a) e b) Irrational Number c) π d) e CK-1 Algebra II with Trigonometry Concepts

3 1. Algebraic Properties 1. commutative. distributive. inverse 4. associative 5. identity 6. inverse a) = 0 b) 6 0 = 0 c) both are valid way to simplify an expression ( 9) ( 4) 0. (6 5) (9) 4 CK-1 Algebra II with Trigonometry Concepts

4 1.4 Evaluating Algebraic Expressions and Equations yes 11. no 1. yes 1. no 14. no 15. yes no 1. yes. yes. no or.5 4. There are at most two solutions to an equation where the largest exponent is = -1 and -4() = -1. The sum is the opposite sign of the coefficient in front of the y-term and the product is the same as the constant (last number) in the equation. CK-1 Algebra II with Trigonometry Concepts 4

5 1.5 Simplifying Algebraic Expressions 1. 5( b d) ( c ) 4g 1 5u 4u cannot be simplified p q p 7x n 17n ( a ) 10. ( x 5) 11. 7( d 1) 1. ( x 8y 7) 1. b(b 5) m m ( 6m 11) 4 y ( y y ) CK-1 Algebra II with Trigonometry Concepts 5

6 1.6 Solving Algebraic Equations for a Variable y x 9 c d f g 5 5 x 15y 5 m n n m 6 5 P 7. w l F C 9 y 4 y 46 y 5 y C cm ft h SA r SA or r r r V r CK-1 Algebra II with Trigonometry Concepts 6

7 1.7 Solving One-Step Equations 1. x =. r = -4. s = 6 4. k = 8 5. m = n = 9 7. y = d = 4 9. p = u = 19 7 or a = 45 6 or b = 8 1. w = b = 15. q = t = x = 18. g = z = k = -45 CK-1 Algebra II with Trigonometry Concepts 7

8 1.8 Solving Two-Step Equations 1. x = -6. x =. x = x = x = - 6. x = x = 0 8. x = x = x = x = x = x = x = x = x = 17. x = or x = 5 8 or x = will vary. Chances are, if students do not like to deal with fractions, they will prefer the LCD method. CK-1 Algebra II with Trigonometry Concepts 8

9 1.9 Solving Multi-Step Equations 1. x = 1. x = -1. x = x = 4 5. x = - 6. x = x = 1 8. x = 5 9. x = x = 17 5 or x = x = - 1. x = x = x = - CK-1 Algebra II with Trigonometry Concepts 9

10 1.10 Solving Basic Inequalities 1. x> -11. x 7. x < x 4 5. x > - 6. x< x x x > x> x 8 1. x< 1 1. x > x > x 16. x> 17. x -7 CK-1 Algebra II with Trigonometry Concepts 10

11 1.11 Solving Multi-Step Inequalities 1. yes. no. no 4. x 1 5. x > x > 7. x x 9. x x< x < x x 14. Even though the x s are on different sides these inequalities are the same. These two problems show us why we need to flip the inequality sign when dividing or multiplying by a negative. 15. The x terms end up canceling out and we are left with -7 > 9 which is an untrue statement. This means there is no solution to this inequality. CK-1 Algebra II with Trigonometry Concepts 11

12 1.1 Compound Inequalities 1. ans ans ans x 1 or x < x < x < x < x x > - or x < x < x > 1 or x - 1. x 48 or x < x < < x < will vary. Students should come up with an or inequality where the solutions are the same number, but going in opposite directions, such as x 1 or x 5 9. Another possibility would be an or inequality where the solutions overlap and continue, such as x 1 or x 1 9. CK-1 Algebra II with Trigonometry Concepts 1

13 1.1 Solving Absolute Value Equations 1. no. no. yes 4. x = 5, x = 9, 9 6. x = -6, x = 1, 9 8. x = 6, x = 5, 10. x = 15, 11. x = 8, x = -5, x = 14. There is only one solution for this absolute value equation because zero does not have a negative. 15. An absolute value equation would have no solution if it is set equal to a negative number. may vary. CK-1 Algebra II with Trigonometry Concepts 1

14 1.14 Solving Absolute Value Inequalities 1. no. yes. yes 4. x> 6 or x < x 5 6. x 5 or x < x < 4 8. x> 6 or x < 9. 4 x x > 8 or x < x x 9 or x 1. x> a or x < a x x a CK-1 Algebra II with Trigonometry Concepts 14

15 1.15 Unit Conversion ft.. 4 c. 100,000 cm 4. 8 pt ,9 cm gal in pt lbs cm c oz. bittersweet, 6 oz. semi-sweet CK-1 Algebra II with Trigonometry Concepts 15

16 1.16 Using Algebraic Models hrs mi. 9, , 5. 4 weeks, but in the last week, you will only have to pay $5. 6. width = 0 ft, length = 40 ft bars dozen ft , 46, 48 CK-1 Algebra II with Trigonometry Concepts 16