Question

1. solve the rational inequality \\(\frac{-5}{x+10} < 0\\). express the answer in interval form.

\\(\bigcirc\\) \\((-\infty, -10)\\)

\\(\bigcirc\\) \\((10, \infty)\\)

\\(\bigcirc\\) \\((-10, \infty)\\)

\\(\bigcirc\\) \\((-\infty, 10)\\)

Explanation:

🆕 New Concept Discovered: Rational Inequalities
Finding where a fraction is positive or negative

Step 1: Analyze the sign of the fraction

We want to find when the rational expression is strictly less than zero (negative):

$$ \frac{-5}{x+10} < 0 $$

A fraction is negative when its numerator and denominator have opposite signs.

The numerator is \(-5\), which is a constant negative number. For the entire fraction to be negative, the denominator must be positive:

$$ \text{Since } \text{numerator} < 0, \text{ we must have } \text{denominator} > 0 $$

Step 2: Solve for x

Set the denominator to be greater than zero:

$$ x + 10 > 0 $$

Subtract \(10\) from both sides:

$$ x > -10 $$

Step 3: Write in interval notation

The inequality \(x > -10\) represents all real numbers strictly greater than \(-10\).

In interval notation, this is written as:

$$ (-10, \infty) $$

Answer:

$$ (-10, \infty) $$