he given equation is either linear or equivalent to a linear equation. Solve the equation. (If there is no solution, enter No solution. If all real numbers are solutions, enter REALS.) \[ \frac{4}{x}=\frac{5}{4 x}+1 \] Need Help? Read it Watch it

To solve the equation: \[ \frac{4}{x} = \frac{5}{4x} + 1 \] follow these steps: 1. **Identify the Restriction:** - Since \( x \) appears in the denominator, \( x \neq 0 \) to avoid division by zero. 2. **Find a Common Denominator:** - The denominators are \( x \) and \( 4x \). - The least common denominator (LCD) is \( 4x \). 3. **Multiply Both Sides by the LCD (\( 4x \)):** \[ 4x \cdot \frac{4}{x} = 4x \cdot \frac{5}{4x} + 4x \cdot 1 \] Simplifying each term: - \( 4x \cdot \frac{4}{x} = 16 \) (since \( x \) cancels out) - \( 4x \cdot \frac{5}{4x} = 5 \) (since \( 4x \) cancels out) - \( 4x \cdot 1 = 4x \) So, the equation becomes: \[ 16 = 5 + 4x \] 4. **Solve for \( x \):** - Subtract 5 from both sides: \[ 16 - 5 = 4x \Rightarrow 11 = 4x \] - Divide both sides by 4: \[ x = \frac{11}{4} \] 5. **Check the Solution:** Substitute \( x = \frac{11}{4} \) back into the original equation to verify: - Left Side: \[ \frac{4}{x} = \frac{4}{\frac{11}{4}} = \frac{16}{11} \] - Right Side: \[ \frac{5}{4x} + 1 = \frac{5}{4 \cdot \frac{11}{4}} + 1 = \frac{5}{11} + 1 = \frac{16}{11} \] Both sides are equal, confirming the solution is correct. **Final Answer:** \( \boxed{\dfrac{11}{4}} \)