Adding and subtracting rational expressions Perform the indicated operations. a. [latex] \frac{x}{x-1}+\frac{2 x+3}{x^{2}-1}[/latex] b. [latex] \frac{x}{x^{2}+6 x+9}-\frac{x-3}{x^{2}+5 x+6}[/latex]

a. [latex] \frac{x}{x-1}+\frac{2 x+3}{x^{2}-1}=\frac{x}{x-1}+\frac{2 x+3}{(x-1)(x+1)}[/latex] Factor denominators completely [latex] =\frac{x(x+1)}{(x-1)(x+1)}+\frac{2 x+3}{(x-1)(x+1)}[/latex] Build up using the LCD (x – 1)(x + 1). [latex] =\frac{x^{2}+x+2 x+3}{(x-1)(x+1)}[/latex] Add the fractions. [latex] =\frac{x^{2}+3 x+3}{(x-1)(x+1)}[/latex] Simplify the numerator. b. [latex] \frac{x}{x^{2}+6 x+9}-\frac{x-3}{x^{2}+5 x+6}=\frac{x}{(x+3)^{2}}-\frac{x-3}{(x+2)(x+3)}[/latex] [latex] \begin{aligned} &=\frac{x(x+2)}{(x+3)^{2}(x+2)}-\frac{(x-3)(x+3)}{(x+2)(x+3)(x+3)} \\ &=\frac{x^{2}+2 x}{(x+3)^{2}(x+2)}-\frac{x^{2}-9}{(x+3)^{2}(x+2)} \\ &=\frac{2 x+9}{(x+3)^{2}(x+2)} \end{aligned} [/latex]

Question B.5.6: Adding and subtracting rational expressions Perform the indi...

Precalculus Functions and Graphs

Adding and subtracting rational expressions

Perform the indicated operations.

a. $\frac{x}{x-1}+\frac{2 x+3}{x^{2}-1}$ b. $\frac{x}{x^{2}+6 x+9}-\frac{x-3}{x^{2}+5 x+6}$

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a. $\frac{x}{x-1}+\frac{2 x+3}{x^{2}-1}=\frac{x}{x-1}+\frac{2 x+3}{(x-1)(x+1)}$ Factor denominators completely

$=\frac{x(x+1)}{(x-1)(x+1)}+\frac{2 x+3}{(x-1)(x+1)}$ Build up using the LCD (x – 1)(x + 1).

$=\frac{x^{2}+x+2 x+3}{(x-1)(x+1)}$ Add the fractions.

$=\frac{x^{2}+3 x+3}{(x-1)(x+1)}$ Simplify the numerator.

b. $\frac{x}{x^{2}+6 x+9}-\frac{x-3}{x^{2}+5 x+6}=\frac{x}{(x+3)^{2}}-\frac{x-3}{(x+2)(x+3)}$