Question

seatwork perform the indicated operations and simplify if possible.

1. $\frac{3x - 1}{x}+\frac{2x + 2}{x + 1}$
2. $\frac{2x - 4}{x^{2}-4}-\frac{3}{x + 2}$
3. $\frac{x + 2}{x - 3}+\frac{2x - 1}{x + 4}$

Explanation:

Step1: Find common denominator for 1.

The common denominator of $x$ and $x + 1$ is $x(x + 1)$.
$\frac{(3x - 1)(x + 1)}{x(x + 1)}+\frac{(2x + 2)x}{x(x + 1)}$

Step2: Expand numerators for 1.

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

Step3: Combine like - terms for 1.

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

Step4: Factor the numerator for 1 (if possible).

$5x^{2}+4x - 1=(5x - 1)(x + 1)$
So the simplified form is $\frac{5x - 1}{x}$

Step5: Factor the denominator for 2.

$x^{2}-4=(x + 2)(x - 2)$
The expression becomes $\frac{2x - 4}{(x + 2)(x - 2)}-\frac{3}{x + 2}$

Step6: Rewrite with common denominator for 2.

The common denominator is $(x + 2)(x - 2)$.
$\frac{2x - 4-3(x - 2)}{(x + 2)(x - 2)}$

Step7: Expand numerator for 2.

$2x - 4-3(x - 2)=2x - 4-3x + 6=-x + 2$
So the simplified form is $\frac{-x + 2}{(x + 2)(x - 2)}=-\frac{1}{x + 2}$

Step8: Find common denominator for 3.

The common denominator of $x - 3$ and $x + 4$ is $(x - 3)(x + 4)$.
$\frac{(x + 2)(x + 4)}{(x - 3)(x + 4)}+\frac{(2x - 1)(x - 3)}{(x - 3)(x + 4)}$

Step9: Expand numerators for 3.

$(x + 2)(x + 4)=x^{2}+4x+2x + 8=x^{2}+6x + 8$ and $(2x - 1)(x - 3)=2x^{2}-6x - x + 3=2x^{2}-7x + 3$
$\frac{x^{2}+6x + 8+2x^{2}-7x + 3}{(x - 3)(x + 4)}$

Step10: Combine like - terms for 3.

$\frac{x^{2}+2x^{2}+6x-7x + 8 + 3}{(x - 3)(x + 4)}=\frac{3x^{2}-x + 11}{(x - 3)(x + 4)}$

Answer:

1. $\frac{5x - 1}{x}$
2. $-\frac{1}{x + 2}$
3. $\frac{3x^{2}-x + 11}{(x - 3)(x + 4)}$