Lesson 6. Diana Pell. Monday, March 17. Section 4.1: Solve Linear Inequalities Using Properties of Inequality

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1 Lesson 6 Diana Pell Monday, March 17 Section 4.1: Solve Linear Inequalities Using Properties of Inequality Example 1. Solve each inequality. Graph the solution set and write it using interval notation. a) 2x 9 10x 3 + 4x + 12 b) 7 > t + 1 1

2 c) 7 < 10 9 s + 2 d) 2 3 (x + 2) > 4 5 (x 3) 2

3 e) (You try!) 3 2 (x + 2) 3 5 (x 3) f) 3a 4 5 > 3a

4 g) (You try!) 4d 5 10 > 2(2d 3) 10 h) 1 2a 2(1 a) 4

5 i) (You try!) 8n (4n 2) Section 4.2: Solving Compound Inequalities Solve Compound Inequalities Containing the Word And. Example 2. Solve x + 3 2x 1 and 3x 2 < 5x 4. Graph the solution set and write it using interval notation. 5

6 Example 3. Solve 2x+3 < 4x+2 and 3x+1 < 5x+3. Graph the solution set and write it using interval notation. Example 4. Solve x 1 > 3 and 2x < 8, if possible. 6

7 Solve Double Linear Inequalities. Example 5. Compound inequality: 3 2x + 5 and 2x + 5 < 7. Double inequality: 3 2x + 5 < 7. Example 6. Solve 5 3x 8 7. Graph the solution set and write it using interval notation. 7

8 Solve Compound Inequalities Containing the Word Or. Example 7. Solve x 3 > 2 or (x 2) > 3. Graph the solution set and 3 write it using interval notation. Example 8. Solve x > 2 or 3(x 2) > 0. Graph the solution set and 2 write it using interval notation. 8

9 Example 9. Solve x or x > 0. Graph the solution set and write it using interval notation. Example 10. Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. a) x 3 x 4 > 1 6 or x

10 b) 3(x ) 7 and 2(x + 2) 2 c) 2 3 x + 1 > 9 and 3 4 x 1 > 10 10

11 Section 4.3: Solving Absolute Value Equations and Inequalities Example 11. Solve x = 3 Example 12. Solve. a) 3x 2 = 5 11

12 b) 10 x = 40 c) x = 1 2 d) x 4 1 = 3 12

13 e) 2x 3 = 7 f) = 10 13

14 g) x 5 4 = 4 h) x = 1 14

15 Example 13. Let f(x) = x + 4. For what value(s) of x is f(x) = 20? Example 14. Solve: 5x + 3 = 3x

16 Solve Inequalities of the Form x < k Example 15. Solve x < 5 and graph the solution set. Example 16. Solve 2x 3 < 9 and graph the solution set. Example 17. Solve 4x 5 < 2 and graph the solution set. 16

17 Solve Inequalities of the Form x > k Example 18. Solve x > 5 and graph the solution set. Example 19. Solve 3 x 5 6 and graph the solution set. 17

18 Example 20. Solve 2 x 4 1 and graph the solution set. Example 21. Solve 6 < 2 3 x 2 3 and graph the solution set. 18

19 Example 22. Solve 3 < 3 4 x and graph the solution set. 19

20 Section 4.4: Linear Inequalities in Two Variables Exercise 1. Graph each inequality. a) y > 3x

21 b) (You Try!) y > 2x 4. c) 2x 3y 6 21

22 d) (You Try!) 3x 2y 12 e) y < 2x 22

23 Section 4.5: Systems of Linear Inequalities Exercise 2. Graph the solution set of each system of inequalities on a rectangular coordinate system. a) { y x + 1 2x y > 2 23

24 b) x 1 y x 4x + 5y < 20 24

25 Exercise 3. A homeowner has a budget of $300 to $600 for trees and bushes to landscape his yard. After shopping, he finds that trees cost $150 each and bushes cost $75 each. What combination of trees and bushes can he afford to buy? Let x = the number of trees purchased and y = the number of bushes purchased. 25

26 Section 5.1: Exponents Properties of Exponents Let a, b R and r, s Z 1) a r a s = a r+s x 11 x 5 2) (a r ) s = a r s (x 11 ) 5 3) (ab) r = a r b r (xy) 3 4) a r = 1 provided that a 0 and r Z+ ar 2 3 5) ( a b ) r = a r b r, b 0 ( x 3) 2 6) ar a s = ar s x 5 x 3 7) a 1 = a and a 0 = 1 (a 0) 26

27 8) ( ) n x ( y ) n. = y x a) ( ) b) ( y 2 x 3 ) 3 c) ( a 2 b 3 a 2 a 3 b 4 ) 3 27

28 ( 2x 2 d) 3y 3 ) 4 Exercise 4. Evaluate each of the following. a) ( 4) 2 b) 4 2 c) ( 4) 2 d) ( ) Exercise 5. Use the properties of exponents to simplify each of the following as much as possible. a) x 5 x 4 b) (2 3 ) 2 c) ( 23 x2 ) 3 d) 3a 2 (2a 4 ) 28

29 Exercise 6. Write each of the following with positive exponents. Then simplify as much as possible. a) 3 2 b) ( 2) 5 c) ( ) d) ( ) ( ) Exercise 7. Simplify each expression. Write all answers with positive exponents only. a) x 4 x 7 b) (a 4 b 3 ) 3 c) (3y 5 ) 2 (2y 4 ) 3 d) ( ) ( ) ( ) x 3 8 x 5 9 x8 29

30 e) (4x 4 y 9 ) 2 (5x 4 y 3 ) 2 Exercise 8. Simplify each expression. Write all answers with positive exponents only. a) a5 a 2 b) t 8 t 5 c) ( x 7 x 4 ) 5 d) (x 4 ) 3 (x 3 ) 4 x 10 e) (6x 3 y 5 ) 2 (3x 4 y 3 ) 4 f) ( x 8 y 3 x 5 y 6 ) 1 30

31 Skip Section 5.2 Section 5.3: Polynomials and Polynomial Functions Definition 23. A term, or monomial, is a constant or the product of a constant and one or more variables raised to whole-number exponent. Exercise 9. The following are monomials (or terms): 14 3x 2 y 2 3 ab2 c 2x Definition 24. A polynomial is any finite sum of terms. Exercise 10. The following are polynomials: 2x 2 + 6x 3 5x 2 y + 2xy 4a 5b + 6c Definition 25. The degree of a polynomial with one variable is the highest power to which the variable is raised in any one term. Addition and Subtraction of Polynomials Exercise 11. Add: ( 1 4 m4 + 1 ) ( 3 2 m3 + 4 m4 7 ) 3 m3. Exercise 12. Subtract 4x 2 9x + 1 from 3x 2 + 5x 2. 31

32 Exercise 13. Simplify (2x + 3) [(3x + 1) (x 7)]. Exercise 14. Simplify 6a { 2a 6[2a + 3(a 1) 6]}. 32

33 Section 5.4: Multiplying Polynomials Exercise 15. Multiply: 1. (3x 2 )(6x 3 ) 2. 2ab(3a 3 b 2a 2 b + 4b 2 ) 3. (3x + 2)(4x + 9) 4. ( 6ab 2 1 ) ( 3ab ) (2a + b)(3a 2 4ab b 2 ) 33

34 Section 5.4 Continued Exercise 16. Multiply: 5cd(c + 6d)(3c 8d) Special Products (x + y) 2 = x 2 + 2xy + y 2 (x y) 2 = x 2 2xy + y 2 Exercise 17. Multiply. a) (5c + 3d) 2 34

35 b) ( ) a4 b 2 c) [(5x + y) + 4] 2 d) [12 + (c d)][12 (c d)] 35

36 Exercise 18. If f(x) = x 2 + 9x 5, find f(a + 4). Exercise 19. If f(x) = x 2 6x + 1, find f(a 8). Exercise 20. Simplify: (5x 4) 2 (x 7)(x + 1) 36

37 Section 5.5: Grouping The Greatest Common Factor and Factoring by The greatest common factor is the largest factor that is common to all terms of the expression. Example 26. Find the GCF of 6a 2 b 3 c, 9a 3 b 2 c, and 18a 4 c 3. Example 27. Factor. a) 16y y b) 25a 3 b 15ab 3 c) 3xy 2 z 3 + 6xyz 3 + 3xz 2 37

38 d) 9x + 16 Example 28. Factor out 1 from n 3 + 2n 2 8 Example 29. Factor. a) x(x + 1) + y(x + 1) b) a(x y + z) b(x y + z) + 3(x y + z) 38

39 c) 2m 2n + mn n 2 d) 7r 7s + rs s 2 e) y 3 + 3y 2 + y + 3 f) x 2 bx x + b 39

40 g) 5x x 2 4x h) 3x 3 y 4x 2 y 2 6x 2 y + 8xy 2 Section 5.6:Factoring Trinomials Multiply: (x + 8)(x 6) 40

41 Exercise 21. Factor each trinomial, if possible. a) n n b) x x + 24 c) x x

42 d) a a + 40 e) 5x 2 + 7x + 2 f) 8t 2 + t

43 g) d d h) 3p 2 4p 4 i) 2q 2 17q 9 43

44 j) 2x 2 y 2 + 4xy 3 30y 4 k) 3a 2 b 2 + 6ab 3 105b 4 l) 7t 2 15t

45 m) 15x xy + 60y 2 n) 6x 2 57xy 72y 2 o) 6y x 2 y 3 + 6x 4 y 3 45

46 p) 4b + 11a 2 b + 6a 4 b Section 5.7: The Difference of Two Squares; the Sum and Difference of Two Cubes Difference of Squares x 2 y 2 = (x y)(x + y) Exercise 22. Factor each expression a) x 2 16 b) 25x 2 36 c) 100w 4 9z 4 46

47 d) 75x 2 3 e) x 4 1 f) a 4 81 g) (x + y) 4 z 4 h) 2x 4 y 32y i) 3a 4 3 j) 2x 4 y 32y Factor the Sum and Difference of Two Cubes 47

48 x 3 + y 3 = (x + y)(x 2 xy + y 2 ) x 3 y 3 = (x y)(x 2 + xy + y 2 ) Exercise 23. Factor each expression a) a b) p c) 27a 3 64b 6 d) a 3 (c + d) 3 e) (p + q) 3 r 3 f) x

49 g) 1 x 6 h) 2a a 2 Exercise 24. Factor each expression completely. a) 60q 2 r 2 s qr 2 s 4 18r 2 s 4 b) ax 2 2axy + ay 2 x 2 + 2xy y 2 c) x4 y 40 d) 8(4 a 2 ) x 3 (4 a 2 ) 49

50 e) (3z + 2) 2 12(3z + 2) + 36 Section 5.9: Solving Equations by Factoring Exercise 25. Solve each equation. a) 2y(4y + 3) = 9 b) x2 9 = 8 9 x

51 c) b 3 5b 2 9b + 45 = 0 d) x2 (6x+37) 35 = x 51