2. Review Topics (Ch R & 1 in College Algebra Book) Exponents & Radical Expressions (P. 21-25 and P. 72-77) Complex Numbers (P. 109 – 114) Factoring (p. 49 – 55) Quadratic Equations (P. 97 – 105) Rational Expressions (P. 61 – 69) Rational Equations & Clearing Fractions (P. 88 – 91) Radical Equations (P. 118 – 123) 1.5: Solving Inequalities 1.6: Equations and Inequalities involving absolute value

3. Review of Exponents 8 2 =8 8 = 642 4 = 2 2 2 2 = 16 x 2 = x xx 4 = x x x xBase = x Exponent = 2Exponent = 4 Exponents of 1Zero Exponents Anything to the 1 power is itself Anything to the zero power = 1 5 1 = 5 x 1 = x (xy) 1 = xy5 0 = 1 x 0 = 1 (xy) 0 = 1 Negative Exponents 5 -2 = 1/(5 2 ) = 1/25 x -2 = 1/(x 2 ) xy -3 = x/(y 3 ) (xy) -3 = 1/(xy) 3 = 1/(x 3 y 3 ) a -n = 1/a n 1/a -n = a n a -n /a -m = a m /a n

4. Powers with Base 10 10 0 = 1 10 1 = 10 10 2 = 100 10 3 = 1000 10 4 = 10000 The exponent is the same as the The exponent is the same as the number number of 0 ’ s after the 1. of digits after the decimal where 1 is placed 10 0 = 1 10 -1 = 1/10 1 = 1/10 =.1 10 -2 = 1/10 2 = 1/100 =.01 10 -3 = 1/10 3 = 1/1000 =.001 10 -4 = 1/10 4 = 1/10000 =.0001 Scientific Notation uses the concept of powers with base 10. Scientific Notation is of the form: __. ______ x 10 (** Note: Only 1 digit to the left of the decimal) You can change standard numbers to scientific notation You can change scientific notation numbers to standard numbers

5. Scientific Notation Scientific Notation uses the concept of powers with base 10. Scientific Notation is of the form: __. ______ x 10 (** Note: Only 1 digit to the left of the decimal) -2 5321 Changing a number from scientific notation to standard form Step 1: Write the number down without the 10 n part. Step 2: Find the decimal point Step 3: Move the decimal point n places in the ‘ number-line ’ direction of the sign of the exponent. Step 4: Fillin any ‘ empty moving spaces ’ with 0. Changing a number from standard form to scientific notation Step1: Locate the decimal point. Step 2: Move the decimal point so there is 1 digit to the left of the decimal. Step 3: Write new number adding a x 10 n where n is the # of digits moved left adding a x10 -n where n is the #digits moved right 5.321.05321 = 5.321 x 10 -2

6. Raising Quotients to Powers a n b = anbnanbn a -n b = a- n b- n = bnanbnan = b n a Examples:3 2 3 2 9 4 4 2 16 == 2x 3 (2x) 3 8x 3 y y 3 y 3 = = 2x -3 (2x) -3 1 y 3 y 3 y y -3 y -3 (2x) 3 (2x) 3 8x 3 == ==

7. Product Rule a m a n = a (m+n) x 3 x 5 = xxx xxxxx = x 8 x -3 x 5 = xxxxx = x 2 = x 2 xxx 1 x 4 y 3 x -3 y 6 = xxxxyyyyyyyyy = xy 9 xxx 3x 2 y 4 x -5 7x = 3xxyyyy 7x = 21x -2 y 4 = 21y 4 xxxxx x 2

8. Quotient Rule a m = a (m-n) a n 4 3 = 4 4 4 = 4 1 = 4 4 3 = 64 = 8 = 4 4 2 4 4 4 2 16 2 x 5 = xxxxx = x 3 x 5 = x (5-2) = x 3 x 2 xx x 2 15x 2 y 3 = 15 xx yyy = 3y 2 15x 2 y 3 = 3 x -2 y 2 = 3y 2 5x 4 y 5 xxxx y x 2 5x 4 y x 2 3a -2 b 5 = 3 bbbbb bbb = b 8 3a -2 b 5 = a (-2-4) b (5-(-3)) = a -6 b 8 = b 8 9a 4 b -3 9aaaa aa 3a 6 9a 4 b -3 3 3 3a 6

9. Powers to Powers (a m ) n = a mn (a 2 ) 3 a 2 a 2 a 2 = aa aa aa = a 6 (2 4 ) -2 = 1 = 1 = 1 = 1/256 ( 2 4)2 2 4 2 4 16 16 2 8 256 (x 3 ) -2 = x –6 = x 10 = x 4 (x -5 ) 2 x –10 x 6 (2 4 ) -2 = 2 -8 = 1 = 1

10. Products to Powers (ab) n = a n b n (6y) 2 = 6 2 y 2 = 36y 2 (2a 2 b -3 ) 2 = 2 2 a 4 b -6 = 4a 4 = a 4(ab 3 ) 3 4a 3 b 9 4a 3 b 9 b 6 b 15 What about this problem? 5.2 x 10 14 = 5.2/3.8 x 10 9  1.37 x 10 9 3.8 x 10 5 Do you know how to do exponents on the calculator?

11. Square Roots & Cube Roots A number b is a square root of a number a if b 2 = a  25 = 5 since 5 2 = 25 Notice that 25 breaks down into 5 5 So,  25 =  5 5 See a ‘ group of 2 ’ -> bring it outside the radical (square root sign). Example:  200 =  2 100 =  2 10 10 = 10  2 A number b is a cube root of a number a if b 3 = a  8 = 2 since 2 3 = 8 Notice that 8 breaks down into 2 2 2 So,  8 =  2 2 2 See a ‘ group of 3 ’ –> bring it outside the radical (the cube root sign) Example:  200 =  2 100 =  2 10 10 =  2 5 2 5 2 =  2 2 2 5 5 = 2  25 3 3 3 3 3 3 3 3 Note:  -25 is not a real number since no number multiplied by itself will be negative Note:  -8 IS a real number (-2) since -2 -2 -2 = -8 3

12. Nth Root ‘ Sign ’ Examples  16  -16 = 4 or -4 not a real number  -16 4 not a real number Even radicals of negative numbers Are not real numbers.  -32 5 = -2 Odd radicals of negative numbers Have 1 negative root.  32 5 = 2 Odd radicals of positive numbers Have 1 positive root. Even radicals of positive numbers Have 2 roots. The principal root Is positive.

13. Exponent Rules (XY) m = x m y m XYXY m = XmYmXmYm

14. Examples to Work through

15. Product Rule and Quotient Rule Example

16. Some Rules for Simplifying Radical Expressions

17. Example Set 1

18. Example Set 2

19. Example Set 3

20. Operations on Radical Expressions Addition and Subtraction (Combining LIKE Terms) Multiplication and Division Rationalizing the Denominator

21. Radical Operations with Numbers

22. Radical Operations with Variables

23. Multiplying Radicals (FOIL works with Radicals Too!)

24. Rationalizing the Denominator Remove all radicals from the denominator

25. Rationalizing Continued… Multiply by the conjugate

26. Complex Numbers REAL NUMBERS Imaginary Numbers Irrational Numbers ,  8, -  13 Rational Numbers (1/2 –7/11, 7/9,.33 Integers (-2, -1, 0, 1, 2, 3...) Whole Numbers (0,1,2,3,4...) Natural Numbers (1,2,3,4...)

27. Complex Numbers (a + bi) Real Numbers a + bi with b = 0 Imaginary Numbers a + bi with b  0 i =  -1 where i 2 = -1 Irrational Numbers Rational Numbers Integers Whole Numbers Natural Numbers

28. Simplifying Complex Numbers A complex number is simplified if it is in standard form: a + bi Addition & Subtraction) Ex1: (5 – 11i) + (7 + 4i) = 12 – 7i Ex2: (-5 + 7i) – (-11 – 6i) = -5 + 7i +11 + 6i = 6 + 13i Multiplication) Ex3: 4i(3 – 5i) = 12i –20i 2 = 12i –20(-1) = 12i +20 = 20 + 12i Ex4: (7 – 3i) (-2 – 5i) [Use FOIL] -14 –35i +6i +15i 2 -14 –29i +15(-1) -14 –29i –15 -29 –29i

29. Complex Conjugates The complex conjugate of (a + bi) is (a – bi) The complex conjugate of (a – bi) is (a + bi) (a + bi) (a – bi) = a 2 + b 2 Division 7 + 4i 2 – 5i 2 + 5i 14 + 35i + 8i + 20i 2 14 + 43i +20(-1) 2 + 5i 4 + 10i –10i – 25i 2 4 –25(-1) 14 + 43i –20 -6 + 43i -6 43 4 + 25 29 29 29 == = + i=

30. Square Root of a Negative Number  25  4 =  100 = 10  -25  -4 =  (-1)(25)  (-1)(4) =  (i 2 )(25)  (i 2 )(4) = i  25 i  4 = (5i) (2i) = 10i 2 = 10(-1) = -10 Optional Step

31. Practice – Square Root of Negatives

32. Practice – Simplify Imaginary Numbers i 2 = i 3 = i 4 = i 5 = i 6 = -i 1 i i 0 = 1 i 1 = i Another way to calculate i n Divide n by 4. If the remainder is r then i n = i r Example: i 11 = __________ 11/4 = 2 remainder 3 So, i 11 = i 3 = -i

33. Practice – Simplify More Imaginary Numbers

34. Practice – Addition/Subtraction 10 +8i -4 +10i

35. Practice – Complex Conjugates Find complex conjugate. 3i => -4i =>

36. Practice Division w/Complex Conjugates 4__ 2i =

37. Adding & Subtracting Polynomials Combine Like Terms (2x 2 –3x +7) + (3x 2 + 4x – 2) = 5x 2 + x + 5 (5x 2 –6x + 1) – (-5x 2 + 3x – 5) = (5x 2 –6x + 1) + (5x 2 - 3x + 5) = 10x 2 – 9x + 6 Types of Polynomials f(x) = 3Degree 0Constant Function f(x) = 5x –3Degree 1Linear f(x) = x 2 –2x –1Degree 2Quadratic f(x) = 3x 3 + 2x 2 – 6Degree 3Cubic

38. Multiplication of Polynomials Step 1: Using the distributive property, multiply every term in the 1 st polynomial by every term in the 2 nd polynomial Step 2: Combine Like Terms Step 3: Place in Decreasing Order of Exponent 4x 2 (2x 3 + 10x 2 – 2x – 5) = 8x 5 + 40x 4 –8x 3 –20x 2 (x + 5) (2x 3 + 10x 2 – 2x – 5) = 2x 4 + 10x 3 – 2x 2 – 5x + 10x 3 + 50x 2 – 10x – 25 = 2x 4 + 20x 3 + 48x 2 –15x -25

39. Binomial Multiplication with FOIL (2x + 3) (x - 7) F.O.I.L. (First)(Outside)(Inside)(Last) (2x)(x)(2x)(-7)(3)(x)(3)(-7) 2x 2 -14x 3x -21 2x 2 -14x + 3x -21 2x 2 - 11x -21

40. Division by a Monomial 3x 2 + x 5x 3 – 15x 2 x 15x 4x 2 + 8x – 12 5x 2 y + 10xy 2 4x 2 5xy 15A 2 – 8A 2 + 12 12A 5 – 8A 2 + 12 4A 4A

41. Review: Factoring Polynomials To factor a polynomial, follow a similar process. Factor: 3x 4 – 9x 3 +12x 2 3x 2 (x 2 – 3x + 4) To factor a number such as 10, find out ‘ what times what ’ = 10 10 = 5(2) Another Example: Factor 2x(x + 1) + 3 (x + 1) (x + 1)(2x + 3)

42. Solving Polynomial Equations By Factoring Solve the Equation: 2x 2 + x = 0 Step 1: Factorx (2x + 1) = 0 Step 2: Zero Productx = 0 or 2x + 1 = 0 Step 3: Solve for Xx = 0 or x = - ½ Zero Product Property : If AB = 0 then A = 0 or B = 0 Question: Why are there 2 values for x???

43. Factoring Trinomials To factor a trinomial means to find 2 binomials whose product gives you the trinomial back again. Consider the expression: x 2 – 7x + 10 (x – 5) (x – 2) The factored form is: Using FOIL, you can multiply the 2 binomials and see that the product gives you the original trinomial expression. How to find the factors of a trinomial: Step 1: Write down 2 parentheses pairs. Step 2: Do the FIRSTS Step3 : Do the SIGNS Step4: Generate factor pairs for LASTS Step5: Use trial and error and check with FOIL

44. Practice Factor: 1.y 2 + 7y –304. –15a 2 –70a + 120 2. 10x 2 +3x –185. 3m 4 + 6m 3 –27m 2 3.8k 2 + 34k +356. x 2 + 10x + 25

45. Special Types of Factoring Square Minus a Square A 2 – B 2 = (A + B) (A – B) Cube minus Cube and Cube plus a Cube (A 3 – B 3 ) = (A – B) (A 2 + AB + B 2 ) (A 3 + B 3 ) = (A + B) (A 2 - AB + B 2 ) Perfect Squares A 2 + 2AB + B 2 = (A + B) 2 A 2 – 2AB + B 2 = (A – B) 2

46. Quadratic Equations General Form of Quadratic Equation ax 2 + bx + c = 0 a, b, c are real numbers & a  0 A quadratic Equation: x 2 – 7x + 10 = 0a = _____ b = _____ c = ______ Methods & Tools for Solving Quadratic Equations 1.Factor 2.Apply zero product principle (If AB = 0 then A = 0 or B = 0) 3.Square root method 4.Completing the Square 5.Quadratic Formula Example1: Example 2: x 2 – 7x + 10 = 04x 2 – 2x = 0 (x – 5) (x – 2) = 02x (2x –1) = 0 x – 5 = 0 or x – 2 = 02x=0 or 2x-1=0 + 5 + 5 + 2 + 2 2 2 +1 +1 2x=1 x = 5 or x = 2 x = 0 or x=1/2 1-7 10

47. Square Root Method If u 2 = d then u =  d or u = -  d. If u 2 = d then u = +  d Solving a Quadratic Equation with the Square Root Method Example 1:Example 2: 4x 2 = 20(x – 2) 2 = 6 4 x – 2 = +  6 x 2 = 5 + 2 + 2 x = +  5 x = 2 +  6 So, x =  5 or -  5 So, x = 2 +  6 or 2 -  6

48. Completing the Square If x 2 + bx is a binomial then by adding b 2 which is the square of half 2 the coefficient of x, a perfect square trinomial results: x2 + bx + b 2 = x + b 2 2 2 Solving a quadratic equation with ‘ completing the square ’ method. Example: Step1: Isolate the Binomial x 2 - 6x + 2 = 0 -2 -2 Step 2: Find ½ the coefficient of x (-3 ) x 2 - 6x = -2 and square it (9) & add to both sides. x 2 - 6x + 9 = -2 + 9 (x – 3) 2 = 7 x – 3 = +  7 x = (3 +  7 ) or (3 -  7 ) Note: If the coefficient of x 2 is not 1 you must divide by the coefficient of x 2 before completing the square. ex: 3x 2 – 2x –4 = 0 (Must divide by 3 before taking ½ coefficient of x) Step 3: Apply square root method (Example 1)

49. (Completing the Square – Example 2) 2x 2 +4x – 1 = 0 2x 2 +4x – 1 = 0 (x + 1) (x + 1) = 3/2 2 2 2 2 (x + 1) 2 = 3/2 x 2 +2x – 1/2 = 0 (x 2 +2x ) = ½ √(x + 1) 2 = √3/2 (x 2 +2x + 1 ) = 1/2 + 1 x + 1 = +/- √6/2 x = √6/2 – 1 or - √6/2 - 1 Step 1: Check the coefficient of the x 2 term. If 1 goto step 2 If not 1, divide both sides by the coefficient of the x 2 term. Step 2: Calculate the value of : (b/2) 2 [In this example: (2/2) 2 = (1) 2 = 1] Step 3: Isolate the binomial by grouping the x 2 and x term together, then add (b/2) 2 to both sides of he equation. Step 4: Factor & apply square root method

50. Quadratic Formula General Form of Quadratic Equation: ax 2 + bx + c = 0 Quadratic Formula: x = -b +  b 2 – 4ac discriminant:  b 2 – 4ac 2a if 0, one real solution if >0, two unequal real solutions if <0, imaginary solutions Solving a quadratic equation with the ‘ Quadratic Formula ’ 2x 2 – 6x + 1= 0a = ______b = ______c = _______ x = - (-6) +  (-6) 2 – 4(2)(1) 2(2) = 6 +  36 –8 4 = 6 +  28 = 6 + 2  7 = 2 (3 +  7 ) = (3 +  7 ) 4 4 4 2 2-61

51. Solving Higher Degree Equations x 3 = 4x x 3 - 4x = 0 x (x 2 – 4) = 0 x (x – 2)(x + 2) = 0 x = 0 x – 2 = 0 x + 2 = 0 x = 2 x = -2 2x 3 + 2x 2 - 12x = 0 2x (x 2 + x – 6) = 0 2x (x + 3) (x – 2) = 0 2x = 0 or x + 3 = 0 or x – 2 = 0 x = 0 or x = -3 or x = 2

52. Solving By Grouping x 3 – 5x 2 – x + 5 = 0 (x 3 – 5x 2 ) + (-x + 5) = 0 x 2 (x – 5) – 1 (x – 5) = 0 (x – 5)(x 2 – 1) = 0 (x – 5)(x – 1) (x + 1) = 0 x – 5 = 0 or x - 1 = 0 or x + 1 = 0 x = 5 or x = 1 or x = -1

53. Rational Expressions Rational Expression – an expression in which a polynomial is divided by another nonzero polynomial. Examples of rational expressions 4 x 2 x 2x – 5 x – 5 Domain = {x | x  0} Domain = {x | x  5/2} Domain = {x | x  5}

54. Multiplication and Division of Rational Expressions A C = A9x = 3 B C B3x 2 x 5y – 10 = 5 (y – 2) = 5 = 1 10y - 20 10 (y – 2) 10 2 2z 2 – 3z – 9 = (2z + 3) (z – 3) = 2z + 3 z 2 + 2z – 15 (z + 5) (z – 3) z + 5 A 2 – B 2 = (A + B)(A – B) = (A – B) A + B (A + B)

55. Negation/Multiplying by –1 -y – 2 4y + 8 -= y + 2 4y + 8 OR -y - 2 -4y - 8

56. Examples x 3 – x x + 1 x – 1 x (x 3 – x) (x + 1) x(x – 1) = x (x 2 – 1)(x + 1) x(x – 1) = = x (x + 1) (x – 1)(x + 1) x(x – 1) = (x + 1)(x + 1) = (x + 1) 2 x 2 – 25 x 2 –10x + 25 x 2 + 5x + 4 2x 2 + 8x  = x 2 – 25 2x 2 + 8x x 2 + 5x + 4 x 2 –10x + 25 = (x + 5) (x – 5) 2x(x + 4) (x + 4)(x + 1) (x – 5) (x – 5) = 2x (x + 5) (x + 1)(x – 5)

57. Check Your Understanding Simplify: x 2 –6x –7 x 2 -1 Simplify: 1 3 x - 2 x 2 + x - 6  (x + 1) (x –7) (x + 1) (x – 1) (x – 7) (x – 1) 1 x 2 + x - 6 x – 2 3 1 (x + 3) (x – 2) x – 2 3 (x + 3) 3

58. Addition of Rational Expressions Adding rational expressions is like adding fractions With LIKE denominators: 1 + 2 = 3 8 8 8 x + 3x - 1 = 4x - 1 x + 2 x + 2 x + 2 x + 2(2 + x) (2 + x) 3x 2 + 4x - 4 3x 2 + 4x -4 (3x 2 + 4x – 4) (3x -2)(x + 2) == = 1 (3x – 2)

59. Adding with UN-Like Denominators 3+ 1 4 8 (3) (2) + 1 8 6+ 1 8 7 8 1 + 2 x 2 – 9 x + 3 1 + 2 (x + 3)(x – 3) (x + 3) 1 + 2 (x – 3) (x + 3)(x – 3) 1 + 2(x – 3) 1 + 2x – 6 2x - 5 (x + 3) (x – 3) (x + 3) (x – 3) (x + 3) (x – 3) = =

60. Subtraction of Rational Expressions 2x - x + 1 x 2 – 1 x 2 - 1 To subtract rational expressions: Step 1: Get a Common Denominator Step 2: Combine Fractions DISTRIBUTING the ‘negative sign’ BE CAREFUL!! = 2x – (x + 1) x 2 -1 = x – 1 (x + 1)(x –1) = 1 (x + 1) = 2x – x - 1 x 2 -1

61. Check Your Understanding Simplify: b b-1 2b - 4 b-2 - b b-1 2(b – 2) b-2 - b -b+1 2(b – 2) b-2 + b 2(b – 2) 2(-b+1) 2(b – 2) + b –2b+2 2(b – 2) -b + 2 2(b – 2) = = -1(b – 2) 2(b – 2) = 2

62. Complex Fractions A complex fraction is a rational expression that contains fractions in its numerator, denominator, or both. Examples: 1 5 4 7 x x 2 – 16 1 x - 4 1x1x + 2x22x2 3x3x - 1x21x2 7/20 x x + 4 x + 2 3x - 1

63. Rational Equations 3x = 3 x + 1 = 3 6 = x 2x – 1 x – 2 x - 2 x + 1 (2x – 1) 3x = 3(2x – 1) 3x = 6x – 3 -3x = -3 x = 1 (x - 2) x + 1 = 3 x = 2 (x + 1) 6 = x (x + 1) 6 = x 2 + x x 2 + x – 6 = 0 (x + 3 ) (x - 2 ) = 0 x = -3 or x = 2 Careful! – What do You notice about the answer?

64. Rational Equations Cont… To solve a rational equation: Step 1: Factor all polynomials Step 2: Find the common denominator Step 3: Multiply all terms by the common denominator Step 4: Solve x + 1 - x – 1 = 1 2x 4x 3 (12x) = 6 (x + 1) -3(x – 1) = 4x 6x + 6 –3x + 3 = 4x 3x + 9 = 4x -3x -3x 9 = x

65. Other Rational Equation Examples 3 + 5 = 12 x – 2 x + 2 x 2 - 4 3 + 5 = 12 x – 2 x + 2 (x + 2) (x – 2) (x + 2)(x – 2) 3(x + 2) + 5(x – 2) = 12 3x + 6 + 5x – 10 = 12 8x – 4 = 12 + 4 + 4 8x = 16 x = 2 1 + 1 = 3 x x 2 4 (4x 2 ) 4x + 4 = 3x 2 3x 2 - 4x - 4 = 0 (3x + 2) (x – 2) = 0 3x + 2 = 0 or x – 2 = 0 3x = -2 or x = 2 x = -2/3 or x = 2

66. Check Your Understanding Simplify: x 1 x 2 – 1 1 3 x – 2 x 1 1 2 x(x – 1) x 2 – 1 x(x + 1) Solve 6 1 x 2 3 2 2x – 1 x + 1 2 3 x x – 1 x + 2 x 2 + x - 2 + - +- -= 1 = += 1 x - 1 2(x – 3) x(x – 2) 3 x(x – 1)(x + 1) 4 5 -1/4 1= 1 + 1 F p q Solve for p: Try this one:

67. Solving Radical Equations X 2 = 64 #1 #2 #3 #4

68. Radical Equations Continued… Example 1: x +  26 – 11x = 4  26 – 11x = 4 - x (  26 – 11x) 2 = (4 – x) 2 26 – 11x = (4-x) (4-x) 26 - 11x = 16 –4x –4x +x 2 26 –11x = 16 –8x + x 2 -26 +11x 0 = x 2 + 3x -10 0 = (x - 2) (x + 5) x – 2 = 0 or x + 5 = 0 x = 2 x = -5 Example 2:  3x + 1 –  x + 4 = 1  3x + 1 =  x + 4 + 1 (  3x + 1) 2 = (  x + 4 + 1) 2 3x + 1 = (  x + 4 + 1) (  x + 4 + 1) 3x + 1 = x + 4 +  x + 4 +  x + 4 + 1 3x + 1 = x + 4 + 2  x + 4 + 1 3x + 1 = x + 5 + 2  x + 4 -x -5 -x -5 2x - 4 = 2  x + 4 (2x - 4) 2 = (2  x + 4) 2 4x 2 –16x +16 = 4(x+4) 4x 2 –20x = 0 4x(x –5) = 0, so…4x = 0 or x – 5 = 0 x = 0 or x = 5 4x+16

69. 1.5 Inequality Set & Interval Notation Set Builder Notation {1,5,6}{ }  {6} {x | x > -4}{x | x < 2}{x | -2 < x < 7} x such that x such that x is lessx such that x is greater x is greater than –4 than or equal to 2than –2 and less than or equal to 7 Interval (-4,  ) (- , 2] (-2, 7] Notation Graph -4 2 -2 0 7 Question: How would you write the set of all real numbers? (- ,  ) or R

70. Inequality Example StatementReason 7x + 15 > 13x + 51 [Given] -6x + 15 > 51 [-13x] -6x > 36 [-15] x < 6[Divide by –6, so must ‘ flip ’ the inequality sign Set Notation: {x | x < 6} Interval Notation: (- , 6] Graph: 6

71. Compound Inequality -3 < 2x + 1 < 3 Set Notation: {x | -2 < x < 1} -1 -1 -1 -4 < 2x < 2 Interval Notation: (-2, 1] 2 2 2 Graph: -2 < x < 1 -2 0 1

72. Set Operations and Compound Inequalities Union (  ) – “OR” A  B = {x | x  A or x  B} -4x + 1  9 or 5X+ 3  12 X  -2 or X  -3 Intersection (  ) – “AND” A  B = {x | x  A and x  B} X+ 1  9 and X – 2  3 X  8 and X  5 Set Notation: {x | X  8 and X  5} Interval Notation: (- , 8]  [5,  ) 0 58 [ ] Set Notation: {x | X  -2 or X  -3} Interval Notation: (- , -2]  (- , -3] -2

73. 1.6 Absolute Value Inequality | 2x + 3| > 5 2x + 3 > 5 or -(2x + 3) > 5 2x > 2 -2x - 3 > 5 x > 1 -2x > 8 -2 -2 x < -4 Set Notation: {x | x 1} Interval Notation: (- , -4] or [1,  ) Graph: -4 01

74. Absolute Value Equations | 2x – 3| = 11 2x – 3 = 11 or -(2x – 3) = 11 2x = 14 -2x + 3 = 11 x = 7 -2x = 8 x = -4