Math 209 Team Textbook Solutions Week 3



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Im not sure some of the original problems are written correctly. I put a “CHECK THIS PROBLEM note under the ones I am suspicious of, so make sure they are presented properly. Also, I don't have the exercises, so try to use the exercises to get the answers. The answers I give are correct, but they might not be done the same way they are derived in the exercises.

Show all relevant steps.

Reduce to lowest terms and simplify.

. Section 6.1: Pages 386-387, Exercises 18, 44, 92, and 102

. Section 6.2: Pages 394-395, Exercises 24, 70, and 76

. Section 6.3: Pages 402-403, Exercises 20 and 66

. Section 6.4: Pages 409-410, Exercises 8, 30, 40, 54, 56

. Section 6.5: Page 419, Exercises 50 and 58

. Section 6.6: Pages 424-425, Exercises 30, 42, and 50

Page 386/18

Which numbers cannot be used in place of the variable in the rational expression? See Example 2.

2y + 1

y2 − y − 6

y2 − y − 6 = (y-3)(y+2) which cannot be zero, so y=3 and y=-2 cannot be used

Page 386/44

Reduce the rational expression to lowest terms. Assume the variables represent only numbers for which the denominators are nonzero. See Example 4.

a2 − b2

a − b

= (a+b)(a-b)/(a-b) = a+b

Page 387/92

Reduce the expression to lowest terms.

4x2

2x9

=2*2*x*x= 2/x7

2*x*x*x7

Page 387/102

Reduce the expression to lowest terms.

−2x − 8 = -2(x+4) = -2

x2 + 2x − 8 (x-2)(x+4) x-2

Page 394/24

Perform the indicated operation. See Example 3.

3 * 2a + 2

a2 + a 6

=3*2*(a+1) = 1

6*a*(a+1) a

Page 395/70

Perform the indicated operation.

− 2a2 * 20a

3a2 15a3

= -2*20*a2*a = -40a = -8

3*15*a2*a3 45a3 9a2

Page 395/76

Perform the indicated operation.

w2 − 1 * w − 1

(w − 1)2 w2 + 2w + 1

= (w+1)(w-1)(w-1) = 1

(w-1)(w-1)(w+1)(w+1) w+1

Page 402/20

Build the rational expression into an equivalent rational expression with the indicated denominator. See Example 1.

− 7yt = ?

3x 18 xyt

cross multiplying

?*(3x)= (-7yt)*(18xyt)

divide by 3x to get ?=(-7yt)*(6yt) = -42y2t2 so

− 7yt = -42y2t2

3x 18 xyt

Page 403/66

Find the LCD for the given rational expression, and convert the rational expression into an equivalent rational expression with the LCD as the denominator. See Example 5.

4 , 5x

x − y 2y − 2x

??? NOTE be sure thats a comma, not a times sign. Also, I don't know how your teacher wants you to find the LCD which is 2(y-x), so you have to do those steps yourself. If thats a comma, then the answer is

-8 , 5x

2(y-x) 2(y-x)

Page 409/8

Perform the indicated operation. Reduce the answer to lowest terms. See Example 1.

4 − 1

9 9

= (4-1) = 3 = 1

9 9 3

Page 410/30

Perform the indicated operation. Reduce the answer to lowest terms. See Example 3.

3-a − a − 5

3 3

= (3-a)-(a-5) = 3-a-a+5 = 8-2a

3 3 3

CHECK THIS PROBLEM

Page 410/40

Perform the indicated operation. Reduce the answer to lowest terms. See Example 4.

y + 2y

4

= y + 4*2y = y+8y = 9y

4 4 4

CHECK THIS PROBLEM

Page 410/54

Perform the indicated operation. Reduce the answer to lowest terms. See Examples 5 and 6.

1 − 2

a − 1 a

=1*a – 2*(a-1) = a-2a+2 = 2-a

a(a-1) a(a-1) a(a-1)

CHECK THIS PROBLEM

Page 410/56

Perform the indicated operation. Reduce the answer to lowest terms. See Examples 5 and 6.

3 + 2

x + 1 x − 1

= 3(x-1)+2(x+1) = 3x-3+2x+2 = 5x-1

(x+1)(x-1) (x+1)(x-1) (x+1)(x-1)

CHECK THIS PROBLEM

Page 419/50

Simplify the complex fraction. Reduce the answer to lowest terms.

Simplify each complex fraction. Reduce the answer to lowest terms.

1 + 1

3 y

y - 3

3 y

Combining the top and bottom terms, then multiplying the top and bottom by 3y, then factoring gives

y+3

= 3y = y+3 = y+3 = 1

y2-9 y2-9 (y+3)(y-3) y-3

3y

Page 419/58

Simplify the complex fraction. Reduce the answer to lowest terms.

ab + b 2

4 ab5

___________

a + b

6 a2 b4

Multiplying the top and bottom by 6 a2 b4/(a+b) gives

(ab+b2)(6a2b4) = 6*b*(a+b)*a*a*b4 = 3a

(4ab5)(a+b) 4*a*b*b4(a+b) 2

Page 424/30

Solve the equation. See Example 3.

x = 4

3 x+1

cross multiplying

x(x+1)=12

x^2+x-12=0

(x-3)(x+4)=0

So x=3 or x=-4

PLEASE CHECK Page 424/30 because they might have some trick to solve this! Anyway, that is the right answer.

Page 424/42

Solve the equation. Check for extraneous solutions (show checks).

5 = y + 7 + 1

y − 3 2y − 6

5 = y + 7 + 2y − 6 = y + 7 + 2y − 6 = 3y + 1

y − 3 2y − 6 2y − 6 2y − 6 2(y-3)

5*2(y-3)=(y-3)(3y+1). Divide by (y-3) which wont be zero as long as y is not equal to 3. 10=3y+1. Subtract 1 from both sides, divide by 3 to get

y=9/3=3

So there is no solution! I think you may have this problem written wrong, something is missing under that last 1.

Page 425/50

Solve the equation.

− 3 = x

x −3

Cross multiplying

(-3)(-3)=x*x or 9=x*x

x=3

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