3-7 Absolute Value Equations and Inequalities

Quick Review

Solving an equation or inequality that contains an absolute value expression is similar to solving other equations and inequalities. You will need to write two equations or inequalities using positive and negative values. Then solve the equations.

Example

What is the solution of | x | − 7 = 3 ? vertical line bold italic x vertical line negative 7 equals 3 question mark

| x | − 7 = 3   | x | = 10 Add  7  to each side . x = 10  or  x = − 10 Definition of absolute value table with 2 rows and 1 column , row1 column 1 , table with 2 rows and 4 columns , row1 column 1 , vertical line x vertical line negative 7 , column 2 equals , column 3 3 , column 4 , row2 column 1 , absolute value of x , , column 2 equals , column 3 10 , column 4 cap add , 7 . toeachside . . , end table , row2 column 1 , table with 1 row and 2 columns , row1 column 1 , x equals 10 , or , x equals negative 10 , column 2 cap definitionofabsolutevalue , end table , end table

Exercises

Solve each equation or inequality. If there is no solution, write no solution .

49. | y | = 3 vertical line y vertical line equals 3
50. | n + 2 | = 4 vertical line n plus 2 vertical line equals 4
51. 4 + | r + 2 | = 7 4 plus vertical line r plus 2 vertical line equals 7
52. | x + 3 | = − 2 vertical line x plus 3 vertical line equals negative 2
53. | 5 x | ≤ 15 vertical line 5 x vertical line less than or equal to 15
54. | 3 d + 5 | < − 2 vertical line 3 d plus 5 vertical line less than negative 2
55. | 2 x − 7 | − 1 > 0 vertical line 2 x minus 7 vertical line negative 1 greater than 0
56. 4 | k + 5 | > 8 4 vertical line k plus 5 vertical line greater than 8
57. Manufacturing The ideal length of a certain nail is 20 mm. The actual length can vary from the ideal by at most 0.4 mm. Find the range of acceptable lengths of the nail.

3-8 Unions and Intersections of Sets

Quick Review

The union of two or more sets is the set that contains all elements of the sets. The intersection of two or more sets is the set of elements that are common to all the sets. Disjoint sets have no elements in common.

Example

Student Activities Of 100 students who play sports or take music lessons, 70 students play a sport and 50 students play a sport and take music lessons. How many students only take music lessons?

So, 30 students take only music lessons.

Exercises

58. Given A = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } eh equals left brace 1 comma 2 comma 3 comma 4 comma 5 comma 6 comma 7 comma 8 comma 9 right brace  and B = { 2 , 4 , 6 , 8 } , b equals left brace 2 comma 4 comma 6 comma 8 right brace comma  what is A ∪ B ? eh union b question mark
59. Let P = { 1 , 5 , 7 , 9 , 13 } , R = { 1 , 2 , 3 , 4 , 5 , 6 , 8 } , p equals left brace 1 comma 5 comma 7 comma 9 comma 13 right brace comma r equals left brace 1 comma 2 comma 3 comma 4 comma 5 comma 6 comma 8 right brace comma  and Q = { 1 , 3 , 5 } . q equals left brace 1 comma 3 comma 5 right brace .  Draw a Venn diagram that represents the intersection and union of the sets.
60. Let N = { x | x  is a multiple of  2 } n equals left brace x vertical line x . isamultipleof . 2 right brace  and P = { x | x  is a multiple of  6 } . p equals left brace x vertical line x . isamultipleof . 6 right brace .  Describe the intersection of N and P.
61. Cats There are 15 cats. Ten are striped and 7 are striped and have green eyes. The rest of the cats have green eyes but are not striped. How many cats have green eyes but are not striped?

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