solve the inequality. x-2 over x^2=5x-14 greater or equal to zero?

Question

express the answer in tems of intervals, if possible. may need inf and cup symbols.

Answer 1

I assume you mean to write:

(x-2)/(x²+5x-14) ≥0

First, let us factor the denominator:

(x-2)/((x-2)(x+7)) ≥0

Now, cancel x-2:

1/(x+7) ≥ 0

Which is true iff:

x+7 > 0
x>-7

Of course, we also know that the cancellation at the beginning is only valid if x≠2, since otherwise x-2 = 0. Thus we must have x>-7 and x≠2, so out solution in interval notation is:

(-7, 2) ∪ (2, ∞)