f a similar problem (6)/(x+1)-(5)/(2)=(2)/(3(x+1))

To solve the equation \(\frac{6}{x+1} - \frac{5}{2} = \frac{2}{3(x+1)}\), follow these steps:<br /><br />1. **Find a common denominator**: The common denominator for the fractions on the left side is \(2(x+1)\).<br /><br />2. **Rewrite each term with the common denominator**:<br /> \[<br /> \frac{6}{x+1} = \frac{6 \cdot 2}{2(x+1)} = \frac{12}{2(x+1)}<br /> \]<br /> \[<br /> \frac{5}{2} = \frac{5(x+1)}{2(x+1)} = \frac{5(x+1)}{2(x+1)}<br /> \]<br /> \[<br /> \frac{2}{3(x+1)} = \frac{2 \cdot 2}{6(x+1)} = \frac{4}{6(x+1)}<br /> \]<br /><br />3. **Combine the fractions**:<br /> \[<br /> \frac{12}{2(x+1)} - \frac{5(x+1)}{2(x+1)} = \frac{4}{6(x+1)}<br /> \]<br /><br />4. **Simplify the left side**:<br /> \[<br /> \frac{12 - 5(x+1)}{2(x+1)} = \frac{4}{6(x+1)}<br /> \]<br /> \[<br /> \frac{12 - 5x - 5}{2(x+1)} = \frac{4}{6(x+1)}<br /> \]<br /> \[<br /> \frac{7 - 5x}{2(x+1)} = \frac{4}{6(x+1)}<br /> \]<br /><br />5. **Cross-multiply to eliminate the denominators**:<br /> \[<br /> 6(7 - 5x) = 8<br /> \]<br /><br />6. **Distribute and simplify**:<br /> \[<br /> 42 - 30x = 8<br /> \]<br /><br />7. **Isolate \(x\)**:<br /> \[<br /> 42 - 8 = 30x<br /> \]<br /> \[<br /> 34 = 30x<br /> \]<br /> \[<br /> x = \frac{34}{30}<br /> \]<br /> \[<br /> x = \frac{17}{15}<br /> \]<br /><br />So, the solution to the equation is:<br />\[<br />x = \frac{17}{15}<br />\]