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
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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
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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
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