Inequality, Interval notation help............
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Inequality, Interval notation help............

[From: ] [author: ] [Date: 12-07-04] [Hit: ]
(x - 7) / (x + 17) >= 0x - 7 >= 0x >= 74. (x - 6)^4 (x - 19)^13 / (x - 199) >= 0(x - 6)^4 (x - 19)^13 >= 0(x - 6) is greater than zero,......
(7x − 55) / (x − 8) ≤ 0

A fraction is ≤ 0 when numerator ≤ 0 and denominator > 0 or when numerator ≥ 0 and denominator < 0

(7x − 55) ≤ 0 -----> x ≤ 55/7
(x − 8) > 0 --------> x > 8
No solution

(7x − 55) ≥ 0 -----> x ≥ 55/7
(x − 8) < 0 --------> x < 8
55/7 ≤ x < 8

Combining results: 55/7 ≤ x < 8

Check: http://www.wolframalpha.com/input/?i=sol…
ok

——————————————————————————————

3.

(x − 7) / (x + 17) ≥ 0

(x − 7) ≥ 0 --------> x ≥ 7
(x + 17) > 0 ------> x > −17
x ≥ 7

(x − 7) ≤ 0 --------> x ≤ 7
(x + 17) < 0 ------> x < −17
x < −17

Combining results: x < −17 or x ≥ 7

Check: http://www.wolframalpha.com/input/?i=sol…
ok

——————————————————————————————

4.

(x − 6)^4 (x − 19)^13 / (x − 199) ≥ 0

Note that (x−6)^4 is always ≥ 0
So this fraction is ≥ 0 if (x − 19)^13 ≥ 0 and (x − 199) > 0
or if (x − 19)^13 ≤ 0 and (x − 199) < 0

Also note that (x−19)^13 ≥ 0 when (x−19) ≥ 0 and (x−19)^13 ≤ 0 when (x−19) ≤ 0

(x − 19) ≥ 0 --------> x ≥ 19
(x − 199) > 0 ------> x > 199
x > 199

(x − 19) ≤ 0 --------> x ≤ 19
(x − 199) < 0 ------> x < 199
x ≤ 19

Combining results: x ≤ 19 or x > 199

Check: http://www.wolframalpha.com/input/?i=sol…
ok

-
1.
1 / (x-6) > 5
1 > 5x - 30
31 > 5x
31/5 > x

2.
(x - 7) / (x - 8) <= -6
x - 7 <= -6x + 48
7x <= 55
x <= 55/7

3.
(x - 7) / (x + 17) >= 0
x - 7 >= 0
x >= 7

4.
(x - 6)^4 (x - 19)^13 / (x - 199) >= 0
(x - 6)^4 (x - 19)^13 >= 0
(x - 6) is greater than zero,
(x - 19)^13 >= 0
x >= 19
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