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

Solve the following inequality. Write the answers in interval notation.

1. 1 / (x-6) > 5

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

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

4. (x - 6)^4 (x - 19)^13 / (x - 199) >= 0


thank you for your help!

1/(x−6) > 5

If this were an equation, you would multiply both sides by (x−6). But we cannot do this with inequalities, since we don't know if (x−6) is positive or negative. So there are two ways to proceed.

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Method 1:

Assume (x−6) is positive (x > 6) , then assume (x−6) is negative (x < 6) and proceed accordingly in both cases.

Case 1A: x−6 is positive (x > 6)

Multiply both sides by x−6
1/(x−6) > 5
1 > 5(x−6)
5(x−6) < 1
5x − 30 < 1
5x < 31
x < 31/5

But since x > 6, we get 6 < x < 31/5


Case 1B: x−6 is negative (x < 6)

Multiply both sides by x−6. Since this is negative, we need to change direction of inequality:
1/(x−6) > 5
1 < 5(x−6)
5(x−6) > 1
5x − 30 > 1
5x > 31
x > 31/5

But since x < 6 it cannot also be > 31/5 ---> no solution for this part


Combining results from both cases, we get 6 < x < 31/5

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Method 2:

Just subtract 5 from both sides, use a common denominator, then factor:

1/(x−6) > 5
1/(x−6) − 5 > 0
(1 − 5(x−6))/(x−6) > 0
(−5x + 31)/(x−6) > 0

A fraction is > 0 if both numerator and denominator are > 0, or when both numerator and denominator < 0

(−5x + 31) > 0 -----> x < 31/5
(x−6) > 0 ------------> x > 6
6 < x < 31/5

(−5x + 31) < 0 -----> x > 31/5
(x−6) < 0 ------------> x < 6
no solution

Combining both: 6 < x < 31/5

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

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2.

(x − 7) / (x − 8) ≤ −6
(x − 7) / (x − 8) + 6 ≤ 0
(x − 7 + 6(x − 8)) / (x − 8) ≤ 0