Solve for \( x \) \( x+\frac{4}{x}<5 \)

To solve the inequality \( x + \frac{4}{x} < 5 \), first, rearrange it to get a common expression on one side: 1. Subtract 5 from both sides: \( x + \frac{4}{x} - 5 < 0 \) 2. This can be rewritten to bring everything to one side: \( x - 5 + \frac{4}{x} < 0 \) 3. To clear the fraction, multiply through by \( x \) (assuming \( x \neq 0 \)): \( x^2 - 5x + 4 < 0 \) 4. Next, factor the quadratic: \( (x - 1)(x - 4) < 0 \) 5. The critical points from the factors are \( x = 1 \) and \( x = 4 \). Test the intervals formed by these points: - For \( x < 1 \) (e.g., \( x = 0 \)): \( (0 - 1)(0 - 4) = 4 > 0 \) - For \( 1 < x < 4 \) (e.g., \( x = 2 \)): \( (2 - 1)(2 - 4) = -2 < 0 \) - For \( x > 4 \) (e.g., \( x = 5 \)): \( (5 - 1)(5 - 4) = 4 > 0 \) 6. The solution to the inequality is where the product is negative: \( 1 < x < 4 \) 7. Finally, remember \( x \) must be positive as the original inequality contains \( \frac{4}{x} \). Therefore, the final solution is: \( (1, 4) \)