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L41-Wed-7-Dec-2016-Rev-Apndx-HW40-and-Rev-Cpt-1-for-Final-HW41

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You get no partial credit for the multiple choice questions but you can for the fill in.

In the MC questions, work out the solution, circle the answer on the test sheet, and shade in the bubble on the answer sheet.

In the MC, some of the questions are set up to make it difficult to guess. For example a question might read:

So, you would add –2 + (–3) to get –5. Circle choice C and shade in the bubble on the answer sheet.

Check my web site for the finals and keys from Spring 2003 and Spring 2005. I did not write those but they are good practice for mine.

The final is important because it pulls things all together so you can see where it all fits.

Now, let’s do some review from the Appendix and Chapter 1.

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2 2

2

2

12 2 3 5 5 2 5 2 12 2 3 5 5 2

12 2 3 5 5

12 2 3 512 3 5 1

3 5 1

14

3

6

14

39 5

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Factor: 3 210 14 12x x x

Always factor out the GCF first.

3 2 2

2

5 3 5 32

10 14 12 2 5 7 6

5 6 301 30 2 15 3 10 5 6

73 10 303 10 7

2 5 3 10 6

2 3 25x xx

x x x x x x

ac

b

x x

x x x x

x

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Factor each denominator to find the LCD. Then, convert each fraction into an equivalent fraction but with the LCD as the denominator.

x x + 1• •

x x + 1

2 23 2 3 2 so LCD 1 1

1 1 113 2

1 1 12 13

1 1 1 13 2 2

1 12

1 1

x x xx x x xx x x

x x x xxx

x x x x x xx x

x x xx

x x x

Combine the terms in the numerator, combine the terms in the denominator, then invert and multiply.

2 2

1 1 2 1 21 1 31 22 2 2 21 1 2 2 1 11 1

x x xx xx x

x x xx xx xx x x x

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Find the LCD of all terms and then multiply each term by the LCD to cancel all fractions.

1 2 13 2

LCD is 6

6 6 6

2 2 6 32 4 6 3

2

xx

x xx x

x

Use a u substitution to put this into quadratic form. That is, 2Let u x

2

2 2

2

2

Let

2

2 02 1 0

2 0 1 02 1

2 12 1

uu u

u uu u

u or uu or u

xx or

x

x

xi

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Isolate the absolute value and then use its definition to break it up into two simple equations.

2 3 1 73 1 9

1 31 3 1 3

4 2

xxxx or x

x or x

1. Read.

2. Diagram – see above.

3. Formulas. None other than Amount = Rate x Base

4. Variable: Let x = # gal of 25% solution

5. Other unknowns: Since we are adding 5 gal to the x amount of 25% soln, the amount of final mix will be x + 5.

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6. Equation: acid in 25% solution acid in 10% solution acid in 18% solution

0.25 0.10 5 0.18 50.25 0.5 0.18 1

x xx x

7. Solve: 0.07 0.5

7.143xx

8. We need about 7.143 gal of the 25% soln.

9. Check: acid in 25% solution acid in 10% solution acid in 18% solution

0.25 0.10 5 0.18 52.28575 2.1

7.143 7.874 Clos

143e enough!

Isolate the x in the center section.

6 4 2 82 2 41 22 1

or 2, 1

xx

xx

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Isolate the absolute value term and then look to see which of the two types it is.

6 2 1 42 1 22 1 2

This has form so or2 1 2 or 2 1 2

1 3or2 2

1 3, ,2 2

xxx

u a u a u ax x

x x

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For the variables, divide the exponent by the index. So, for

3 165

we have 3 16

151

x

The quotient, 5, is the power of x outside the radical while the remainder, 1, is the power of x inside the radical. Like wise for the power of y. 25/3 is 8 with remainder 1.

3 3 316 25 3 15 1 24 1 5 8 1 18 2 2x y x x y y x y x y

Simplify each radical first and then combine like radical terms.

3 33 3 3 34 3

3 3 3

3

16 2 2 3 54 2 2 2 3 2 3

2 2 2 2 3 3 2

13 2

Multiply numerator and denominator by conjugate of the denominator. That will give the difference of two perfect squares in the denominator and so the radical will disappear.

1 1 2 3 2 3 2 34 32 3 2 3 2 3

Use the same procedure show above for radicals.

21 1 2 2 2 2

2 2 2 54 14i i i i

i i i i

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5 2 1 32 1 2

1 11 1

2

xxxx

x

Rewrite using rational exponents:

11

613 1 13

35

56212

x x x x x x xx

x

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Convert the square root of a negative into the square root of the positive and them multiply by i.

23 20 3 20

3 2 2 5 4 15 1 2 15

i i

i

.

We have to get rid of the i in the denominator first.

2

2

22 2 2

2

4 2 224 12 1

51 2 Complex for means form

5 5

i i ii i i

i i

i i ii i

i

i a bi

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First, find slope of the line that passes through the points (2, 3) and (1, -5):

2 1

2 1

5 3 8 881 2 1 1

y yymx x x

The perpendicular slope is

negative reciprocal18

m

Now, use y = mx + b to find b.

10 58508

58

y mx b

b

b

b

So, 1 58 8

y x

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Be sure to divide by 2 so the coefficients of the squared terms are 1. Then, complete the square.

2 2

2 2

2 2

2 2

2 2 2 22 2

2 2

1 22 2

2 2

222

2 2 4 8 12 0

2 4 6 0

2 4 6 0

2 2 4 42 4 6 02 2 2 2

2 1 1 4 4 4 6 0

1 1 2 4 6 0

1 2 11

1 2 11

x y

x y x yx y x y

x x y y

x x y y

x x y y

x y

x y

x y

The center is (1, –2) and

the radius is 1 1 .