To solve the rational inequality [tex]\(\frac{-5}{x+10}<0\)[/tex], we need to determine the values of [tex]\(x\)[/tex] for which the given expression is less than zero.
Let's analyze the inequality step-by-step:
1. The expression [tex]\(\frac{-5}{x+10}\)[/tex] will be negative if and only if the denominator [tex]\((x + 10)\)[/tex] is positive, since the numerator [tex]\(-5\)[/tex] is already negative. This is because a negative numerator divided by a positive denominator results in a negative value.
2. To find when [tex]\(x + 10\)[/tex] is positive, we solve the inequality:
[tex]\[
x + 10 > 0
\][/tex]
3. Subtracting 10 from both sides, we get:
[tex]\[
x > -10
\][/tex]
Therefore, the solution to the inequality [tex]\(\frac{-5}{x+10}<0\)[/tex] is all [tex]\(x\)[/tex] such that [tex]\(x > -10\)[/tex].
In interval notation, this solution is expressed as:
[tex]\[
(-\infty, -10)
\][/tex]
Thus, the correct answer is:
[tex]\[
(-\infty, -10)
\][/tex]
