Questions: The rational expression 5/(2x) - 3x/(x+1) is equivalent to .

Transcript text: The rational expression $\frac{5}{2 x}-\frac{3 x}{x+1}$ is equivalent to $\qquad$ .

Solution

Solution Steps

To combine the rational expressions, we need to find a common denominator. The common denominator for the fractions $\frac{5}{2x}$ and $\frac{3x}{x+1}$ is the product of the individual denominators, which is $2x(x+1)$. We then rewrite each fraction with this common denominator and combine them.

Step 1: Identify the Common Denominator

To combine the rational expressions $\frac{5}{2x}$ and $\frac{3x}{x+1}$, we first identify the common denominator. The common denominator is the product of the individual denominators: \[ 2x \cdot (x + 1) = 2x(x + 1) \]

Step 2: Rewrite Each Fraction with the Common Denominator

Next, we rewrite each fraction with the common denominator: \[ \frac{5}{2x} = \frac{5(x + 1)}{2x(x + 1)} \] \[ \frac{3x}{x+1} = \frac{3x \cdot 2x}{2x(x + 1)} = \frac{6x^2}{2x(x + 1)} \]

Step 3: Combine the Fractions

We then combine the fractions: \[ \frac{5(x + 1)}{2x(x + 1)} - \frac{6x^2}{2x(x + 1)} = \frac{5(x + 1) - 6x^2}{2x(x + 1)} \]

Step 4: Simplify the Numerator

Simplify the numerator: \[ 5(x + 1) - 6x^2 = 5x + 5 - 6x^2 \] Thus, the combined expression is: \[ \frac{5x + 5 - 6x^2}{2x(x + 1)} \]