Help with a rational inequality?

Question

x+1
----- >0
x-5

Solve. Put answer in interval notation and use ; to denote union

Answer 1

This will be positive if both the numerator and the denominator are positive OR if both numerator and the denominator are negative. Tha is , if they have the same sign.

Both will be positive if x > -1 AND x > 5, which simply means that x > 5 because -1 > 5. So, if x is in (5, oo)
And both will be negative if x< -1 and x < 5, which simply means that x < -1 because -1 < 5. So, x is in (-1, -oo).

Since both conditions rend the rational fraction positive, it will be positive if x is in (-oo , -1) U (5, 00).

Answer 2

For this to be true, both the numerator and denominator must have the same sign. So either:

x + 1 > 0 AND x - 5 > 0

Or

x + 1 < 0 AND x - 5 < 0

Solve:

x > -1 and x > 5 --> x > -1 (since x > 5 is a part of the solution)
x < -1 and x < 5 --> x < 5 (since x < -1 is a part of the solution)

-1 < x < 5

In interval notation:

(-1, 5)

Answer 3

This inequality is equivalent to (x+1)(x-5) > 0 when x ≠ 5. Therefore, the solution is the union of (∞, -1) and (5, ∞).

Answer 4

Correct answer is (-inf, -1) ; (5, inf)

Yeeehaw was almost correct but change the following lines

x > -1 and x > 5 --> x > 5
x < -1 and x < 5 --> x < -1