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Hi,

1. If 0 < x < 4, then a < 2x + 3 < b. find the sum of a and b.

If x = 0. then a < 2(0) + 3, so a < 3, Assuming it;s an integer, use a = 2.

If x = 4, then 2(4) + 3 < b, so 11 < b, so use 12 for b

a + b = 2 + 12 = 14 <==ANSWER

2. p = 3, q = 1, and r = -2, so p + q + r = 2 <==ANSWER

3. (2x - 4)/5 = 14

2x - 4 = 70

2x = 74

x = 37

(2x - 4)/5 = -14

2x - 4 = -70

2x = -66

x = -33

37 + -33 = 4 <==ANSWER

. . ====-----------

4) √2 + √2 + √2 ... = 2 <==ANSWER

5) 5 + |x - 1| > ½

|x - 1| > -4½

An absolute value is always greater than any negative number.

ANSWER: all real numbers <==ANSWER

. . . . . . . . _____

6) 2 - y - 2√y + 1 = 0

. _____

2√y + 1 = 2 - y

Square both sides.

4(y + 1) = 4 - 4y + y²

4y + 4 = 4 - 4y + y²

y² - 8y = 0

y(y - 8) = 0

y = 0 <==ANSWER

(8 does not check in original equation)

7. x = -1/3 ±√(7)/2

x = -1/3 - √(7)/2 and x = -1/3 + √(7)/2

(x + 1/3 + √(7)/2)(x + 1/3 - √(7)/2) = y

y = x² + 2/3x + 1/9 - 7/4

y = x² + 2/3x - 59/36 <==ANSWER

8. 3 - 4x - x² > 0

x² + 4x - 3 < 0

. . . .__________

-4 ± √4² - 4(1)(-3)

-------------------------- = x

. . . . . 2

. . . ._______

-4 ± √16 + 12

--------------------- = x

. . . . . 2

. . . .__

-2 ± √7 = x

. . . . _ . . . . ._

(-2 - √7, -2 + √7) <==ANSWER

9. a = 3/2c

b = 5/4c

a = 6/5b <==ANSWER

. . . . . _____________

. . . . ./ . . ====___==

10. ³√ N ³√N .³√ N

³√ N ³√ N*N^(1/3)

³√ N ³√ N^(4/3)

³√ N * N^(4/9)

³√ N^(13/9)

N^(13/27) <==ANSWER

. .___ . . . . . . or

²⁷√N¹³ <==ANSWER

11. 2500 - 622 = 1878/2 = 939 discount price

939 + 622 = 1561 <==ANSWER for full price

12. 1/4n² = 1/2n + 3/8

Multiply every term by 8 to eliminate fractions. Then solve.

2n² = 4n + 3

2n² - 4n - 3 = 0

. . . . . .___________

-(-4) ± √(-4)² - 4(2)(-3)

------------------------------- = n

. . . . . 2(2)

. . . ._______

4 ± √16 + 24

-------------------- = n

. . . . . 4

. . . . ___

4 ± 2√10

-------------- = n

. . 4

. . . ___

2 ± √10

------------ = n <==ANSWER

. . 2

13. . . . __________

. . . . . / . . . x

. . . . ./ ------------------ if x > 0

. . . . / . . . . x - 1

. .\ . / . 1 - ---------

. . .\/ . . . . . . x

Multiply top and bottom both by x to eliminate the little fraction.

. . . . . __________

. . . . . / . . . x²

. . . \ ./ ------------------ =

. . . .\/ . x - (x - 1)

. . . . . __________

. . . . . / . . . x²

. . . \ ./ ------------------ =

. . . .\/ . . . . 1

. ____

√ x² = x <==ANSWER

14. If x $ y = x³ - y, then what is h $ (h $ h)?

(I changed the symbol.)

h $ h = h³ - h

h $ (h $ h) = h $ (h³ - h) = h³ - (h³ - h) = h³ - h³ + h = h <==ANSWER

15. N + D = 20

5N + 10D + 70 = 5D + 10N

5(N + D = 20)

-5N + 5D = -70

5N + 5D = 100

-5N + 5D = -70

-----------------------

10D = 30

D = 3

N = 17

There are 17 nickels and 3 dimes. <==ANSWER

16. . . . . 1

. . x = -----------

. . . . . . 3 + x

x(3 + x) = 1

3x + x² = 1

x² + 3x - 1 = 0

. . . . ____________

-3 ± √(-3)² - 4(1)(-1)

----------------------------- = x

. . . . . 2(1)

. . . . _____

-3 ± √9 +4

------------------- = x

. . .2

. . . . ____

-3 ± √ 13

---------------- = x <==ANSWER

. . .2

17. |3x - |2x + 1|| = 4

3x - |2x + 1| = 4

|2x + 1| = 3x - 4

2x + 1 = 3x - 4 or 2x + 1 = -3x + 4 <= x = 3/5 does not check

x = 5 <==ANSWER (partial)

3x - |2x + 1| = -4

|2x + 1| = 3x + 4

2x + 1 = 3x + 4 or 2x + 1 = -3x - 4 <=x = -3 does not check

x = -1 <==ANSWER

18. a^(2/5) b^(1/6) . . a^(1/10)

. . --------------------- = ------------- <==ANSWER

. . .a^(3/10)b^(1/4) . . b^(1/12)

19. 2 + 7 + 31 = 40, so 2(7)(31) = 434 <==ANSWER

20. √a .³√b .⁴√c = a^(1/2) b^(1/3) c^(1/4) = a^(6/12) b^(4/12) c^(3/12) = ¹²√a⁶b⁴c³

6 + 4 + 3 = 13 <==ANSWER

I hope that helps!! :-)

TWENTY questions and a fair bit of work in each of them?

I feel faint !