5-x / x+4 >0

To solve the inequality \( \frac{5 - x}{x + 4} > 0 \), we need to find the values of \( x \) for which the fraction is positive.

Step 1: Determine when the numerator and denominator are positive or negative.

Numerator: \( 5 - x \)

• \( 5 - x > 0 \) when \( x < 5 \) (since we want the numerator to be positive).

Denominator: \( x + 4 \)

• \( x + 4 > 0 \) when \( x > -4 \) (since we want the denominator to be positive).

Step 2: Analyze the signs of the expression

Next, we will determine the sign of \( \frac{5 - x}{x + 4} \) in the intervals formed by the critical points \( x = -4 \) and \( x = 5 \).

Critical Points:

1. \( x = -4 \) (denominator changes sign)
2. \( x = 5 \) (numerator changes sign)

The intervals to examine are:

• \( (-\infty, -4) \)
• \( (-4, 5) \)
• \( (5, \infty) \)

Step 3: Test each interval

1. Interval: \( (-\infty, -4) \)

   • Choose \( x = -5 \): \[ \frac{5 - (-5)}{-5 + 4} = \frac{10}{-1} = -10 \quad (\text{Negative}) \]

2. Interval: \( (-4, 5) \)

   • Choose \( x = 0 \): \[ \frac{5 - 0}{0 + 4} = \frac{5}{4} \quad (\text{Positive}) \]

3. Interval: \( (5, \infty) \)

   • Choose \( x = 6 \): \[ \frac{5 - 6}{6 + 4} = \frac{-1}{10} \quad (\text{Negative}) \]

Step 4: Compile the results

From this analysis, we find:

• The inequality \( \frac{5 - x}{x + 4} > 0 \) holds true in the interval \( (-4, 5) \).

Step 5: Check endpoints

• At \( x = -4 \): The denominator becomes zero (undefined).
• At \( x = 5 \): The numerator becomes zero (equals zero, not greater than zero).

Final Solution

Thus, the solution for the inequality \( \frac{5 - x}{x + 4} > 0 \) is:

\[ \boxed{(-4, 5)} \]