Section 4.1: Solve Linear Inequalities Using Properties of Inequality

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1 Diana Pell 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 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. Example 3. Solve x 1 > 3 and 2x < 8, if possible. 3

4 Solve Double Linear Inequalities. Example 4. Solve 3 2x + 5 < 7. Graph the solution set and write it using interval notation. Note: Solve 15 < 5x 25. Solve Compound Inequalities Containing the Word Or. Example 5. Solve x 3 > 2 or (x 2) > 3. Graph the solution set and 3 write it using interval notation. 4

5 Example 6. Solve x > 2 or 3(x 2) > 0. Graph the solution set and 2 write it using interval notation. Example 7. Solve x or x > 0. Graph the solution set and write it using interval notation. 5

6 Example 8. 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 b) 3(x ) 7 and 2(x + 2) 2 6

7 Section 4.3: Solving Absolute Value Equations and Inequalities Example 9. Solve x = 3 Example 10. Solve. a) x = 1 2 b) 3x 2 = 5 c) 10 x = 40 7

8 d) 2x 3 = 7 e) = 10 f) x 5 4 = 4 8

9 g) x = 1 Example 11. Let f(x) = x + 4. For what value(s) of x is f(x) = 20? 9

10 Example 12. Solve: 5x + 3 = 3x + 25 Solve Inequalities of the Form x < k For any positive number k and any algebraic expression X: To solve X < k, solve the equivalent double inequality k < X < k. To solve X k, solve the equivalent double inequality k X k. Example 13. Solve x < 5 and graph the solution set. 10

11 Example 14. Solve 2x 3 < 9 and graph the solution set. Example 15. Solve 4x 5 < 2 and graph the solution set. Solve Inequalities of the Form x > k For any positive number k and any algebraic expression X: To solve X > k, solve the equivalent compound inequality X > k or X < k. To solve X k, solve the equivalent compound inequality X k or X k. 11

12 Example 16. Solve x > 5 and graph the solution set. Example 17. Solve 3 x 5 6 and graph the solution set. 12

13 Example 18. Solve 2 x 4 1 and graph the solution set. Example 19. Solve 6 < 2 3 x 2 3 and graph the solution set. 13

14 Example 20. Solve 3 < 3 4 x and graph the solution set. Example 21. Solve x 8 x 1 4 and graph the solution set. 14

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

16 b) (You Try!) y > 2x 4. c) 2x 3y 6 16

17 d) (You Try!) 3x 2y 12 e) y < 2x 17

18 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 18

19 b) x 1 y x 4x + 5y < 20 19

20 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. 20

21 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) 21

22 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 22

23 ( 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 ) 23

24 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 24

25 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 25

26 Section 5.3: Polynomials and Polynomial Functions Definition 22. 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 23. 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 24. 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. 26

27 Section 5.4: Multiplying Polynomials Exercise 13. Multiply: 1. (3x 2 )(6x 3 ) 2. 2ab(3a 3 b 2a 2 b + 4b 2 ) 3. (3x + 2)(4x + 9) 4. (2a + b)(3a 2 4ab b 2 ) Exercise 14. Multiply: 5cd(c + 6d)(3c 8d) 27

28 Special Products (x + y) 2 = x 2 + 2xy + y 2 (x y) 2 = x 2 2xy + y 2 Exercise 15. Multiply. a) (5c + 3d) 2 b) ( ) a4 b 2 c) [(5x + y) + 4] 2 28

29 Exercise 16. If f(x) = x 2 + 9x 5, find f(a + 4). Exercise 17. If f(x) = x 2 6x + 1, find f(a 8). Exercise 18. Simplify: (5x 4) 2 (x 7)(x + 1) 29

30 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 25. Find the GCF of 6a 2 b 3 c, 9a 3 b 2 c, and 18a 4 c 3. Example 26. Factor. a) 16y y b) 3xy 2 z 3 + 6xyz 3 + 3xz 2 Example 27. Factor out 1 from n 3 + 2n

31 Example 28. Factor. a) x(x + 1) + y(x + 1) b) a(x y + z) b(x y + z) + 3(x y + z) c) 2m 2n + mn n 2 d) 7r 7s + rs s 2 31

32 e) y 3 + 3y 2 + y + 3 f) x 2 bx x + b g) 5x x 2 4x 32

33 h) 3x 3 y 4x 2 y 2 6x 2 y + 8xy 2 Section 5.6:Factoring Trinomials Multiply: (x + 8)(x 6) Exercise 19. Factor each trinomial, if possible. a) n n b) x x

34 c) x x + 24 d) 5x 2 + 7x + 2 e) 8t 2 + t f) d d g) 3p 2 4p 4 34

35 h) 2q 2 17q 9 i) 2x 2 y 2 + 4xy 3 30y 4 j) 3a 2 b 2 + 6ab 3 105b 4 k) 7t 2 15t

36 l) 15x xy + 60y 2 m) 6x 2 57xy 72y 2 n) 6y x 2 y 3 + 6x 4 y 3 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 20. Factor each expression a) x 2 16 b) 25x

37 c) 100w 4 9z 4 d) 75x 2 3 e) x 4 1 f) a 4 81 g) (x + y) 4 z 4 h) 2x 4 y 32y Factor the Sum and Difference of Two Cubes x 3 + y 3 = (x + y)(x 2 xy + y 2 ) x 3 y 3 = (x y)(x 2 + xy + y 2 ) Exercise 21. Factor each expression a) a

38 b) p c) 27a 3 64b 6 d) a 3 (c + d) 3 e) (p + q) 3 r 3 f) x 6 64 Exercise 22. Factor each expression completely. a) 60q 2 r 2 s qr 2 s 4 18r 2 s 4 38

39 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 ) e) (3z + 2) 2 12(3z + 2)

40 Section 5.9: Solving Equations by Factoring Exercise 23. Solve each equation. a) 2y(4y + 3) = 9 b) x2 9 = 8 9 x 7 9 c) b 3 5b 2 9b + 45 = 0 40

41 d) x2 (6x+37) 35 = x 41