Answered

Solve the rational inequality [tex]\(\frac{-5}{x+10}\ \textless \ 0\)[/tex]. Express the answer in interval form.

A. [tex]\((-\infty, 10)\)[/tex]
B. [tex]\((10, \infty)\)[/tex]
C. [tex]\((-10, \infty)\)[/tex]
D. [tex]\((-\infty, -10)\)[/tex]



Answer :

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]