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