Question

question 3
use the lcd to combine the two rational expressions.
$\frac{4x}{x + 1} - \frac{x + 1}{4x} = $
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question 4
simplify the rational expression.
$\frac{\frac{4}{x} + \frac{4}{y}}{x} = $
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Explanation:

Question 3

Step1: Find the LCD

The least common denominator (LCD) of $x+1$ and $4x$ is $4x(x+1)$.

Step2: Rewrite fractions with LCD

$$\frac{4x}{x+1} = \frac{4x \cdot 4x}{4x(x+1)} = \frac{16x^2}{4x(x+1)}$$
$$\frac{x+1}{4x} = \frac{(x+1)(x+1)}{4x(x+1)} = \frac{(x+1)^2}{4x(x+1)}$$

Step3: Subtract the fractions

$$\frac{16x^2}{4x(x+1)} - \frac{(x+1)^2}{4x(x+1)} = \frac{16x^2 - (x+1)^2}{4x(x+1)}$$

Step4: Expand and simplify numerator

$$16x^2 - (x^2 + 2x + 1) = 16x^2 - x^2 - 2x - 1 = 15x^2 - 2x - 1$$

Step5: Factor numerator (optional, fully simplified)

$$15x^2 - 2x - 1 = (3x - 1)(5x + 1)$$

Step1: Simplify numerator first

Find LCD of $\frac{4}{x}$ and $\frac{4}{y}$, which is $xy$.
$$\frac{4}{x} + \frac{4}{y} = \frac{4y + 4x}{xy} = \frac{4(x+y)}{xy}$$

Step2: Divide by $x$ (multiply by reciprocal)

$$\frac{\frac{4(x+y)}{xy}}{x} = \frac{4(x+y)}{xy} \cdot \frac{1}{x}$$

Step3: Multiply denominators

$$\frac{4(x+y)}{x^2y}$$

Answer:

$\frac{15x^2 - 2x - 1}{4x(x+1)}$ or $\frac{(3x-1)(5x+1)}{4x(x+1)}$

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Question 4