Simplify. Express your answer as a single fraction in simplest form \( \frac{u}{8 u+1}-\frac{3 u}{u+1} \)
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To simplify the expression \( \frac{u}{8u+1}-\frac{3u}{u+1} \), we first need a common denominator. The denominators are \( 8u + 1 \) and \( u + 1 \). The common denominator is \( (8u + 1)(u + 1) \). Rewriting each fraction with the common denominator gives: \[ \frac{u(u + 1)}{(8u + 1)(u + 1)} - \frac{3u(8u + 1)}{(8u + 1)(u + 1)} \] Now, combine the two fractions: \[ \frac{u(u + 1) - 3u(8u + 1)}{(8u + 1)(u + 1)} \] Now, simplify the numerator: \[ u(u + 1) = u^2 + u \] \[ 3u(8u + 1) = 24u^2 + 3u \] So, the combined numerator becomes: \[ u^2 + u - (24u^2 + 3u) = u^2 + u - 24u^2 - 3u = -23u^2 - 2u \] Hence, our expression is: \[ \frac{-23u^2 - 2u}{(8u + 1)(u + 1)} \] Now, we can factor the numerator: \[ -1(23u^2 + 2u) = -1 \cdot u(23u + 2) \] Putting it all together, the expression simplifies to: \[ \frac{-u(23u + 2)}{(8u + 1)(u + 1)} \] Thus, the final answer is: \[ \frac{-u(23u + 2)}{(8u + 1)(u + 1)} \]
