Simplify (x+6)/(x-1)+(1-x)/(x+3)

$\frac{x + 6}{x - 1} + \frac{1 - x}{x + 3}$

Step 1

To write $\frac{x + 6}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$\frac{x + 6}{x - 1} \cdot \frac{x + 3}{x + 3} + \frac{1 - x}{x + 3}$

Step 2

To write $\frac{1 - x}{x + 3}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{x + 6}{x - 1} \cdot \frac{x + 3}{x + 3} + \frac{1 - x}{x + 3} \cdot \frac{x - 1}{x - 1}$

Step 3

Write each expression with a common denominator of $(x - 1) (x + 3)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x + 6}{x - 1}$ by $\frac{x + 3}{x + 3}$ .

$\frac{(x + 6) (x + 3)}{(x - 1) (x + 3)} + \frac{1 - x}{x + 3} \cdot \frac{x - 1}{x - 1}$

Multiply $\frac{1 - x}{x + 3}$ by $\frac{x - 1}{x - 1}$ .

$\frac{(x + 6) (x + 3)}{(x - 1) (x + 3)} + \frac{(1 - x) (x - 1)}{(x + 3) (x - 1)}$

Reorder the factors of $(x - 1) (x + 3)$ .

$\frac{(x + 6) (x + 3)}{(x + 3) (x - 1)} + \frac{(1 - x) (x - 1)}{(x + 3) (x - 1)}$

Step 4

Combine the numerators over the common denominator.

$\frac{(x + 6) (x + 3) + (1 - x) (x - 1)}{(x + 3) (x - 1)}$

Step 5

Simplify the numerator.

Expand $(x + 6) (x + 3)$ using the FOIL Method.

Apply the distributive property.

$\frac{x (x + 3) + 6 (x + 3) + (1 - x) (x - 1)}{(x + 3) (x - 1)}$

Apply the distributive property.

$\frac{x \cdot x + x \cdot 3 + 6 (x + 3) + (1 - x) (x - 1)}{(x + 3) (x - 1)}$

Apply the distributive property.

$\frac{x \cdot x + x \cdot 3 + 6 x + 6 \cdot 3 + (1 - x) (x - 1)}{(x + 3) (x - 1)}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 3 + 6 x + 6 \cdot 3 + (1 - x) (x - 1)}{(x + 3) (x - 1)}$

Move $3$ to the left of $x$ .

$\frac{x^{2} + 3 \cdot x + 6 x + 6 \cdot 3 + (1 - x) (x - 1)}{(x + 3) (x - 1)}$

Multiply $6$ by $3$ .

$\frac{x^{2} + 3 x + 6 x + 18 + (1 - x) (x - 1)}{(x + 3) (x - 1)}$

Add $3 x$ and $6 x$ .

$\frac{x^{2} + 9 x + 18 + (1 - x) (x - 1)}{(x + 3) (x - 1)}$

Expand $(1 - x) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$\frac{x^{2} + 9 x + 18 + 1 (x - 1) - x (x - 1)}{(x + 3) (x - 1)}$

Apply the distributive property.

$\frac{x^{2} + 9 x + 18 + 1 x + 1 \cdot - 1 - x (x - 1)}{(x + 3) (x - 1)}$

Apply the distributive property.

$\frac{x^{2} + 9 x + 18 + 1 x + 1 \cdot - 1 - x \cdot x - x \cdot - 1}{(x + 3) (x - 1)}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $1$ .

$\frac{x^{2} + 9 x + 18 + x + 1 \cdot - 1 - x \cdot x - x \cdot - 1}{(x + 3) (x - 1)}$

Multiply $- 1$ by $1$ .

$\frac{x^{2} + 9 x + 18 + x - 1 - x \cdot x - x \cdot - 1}{(x + 3) (x - 1)}$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{x^{2} + 9 x + 18 + x - 1 - (x \cdot x) - x \cdot - 1}{(x + 3) (x - 1)}$

Multiply $x$ by $x$ .

$\frac{x^{2} + 9 x + 18 + x - 1 - x^{2} - x \cdot - 1}{(x + 3) (x - 1)}$

Multiply $- x \cdot - 1$ .

Multiply $- 1$ by $- 1$ .

$\frac{x^{2} + 9 x + 18 + x - 1 - x^{2} + 1 x}{(x + 3) (x - 1)}$

Multiply $x$ by $1$ .

$\frac{x^{2} + 9 x + 18 + x - 1 - x^{2} + x}{(x + 3) (x - 1)}$

Add $x$ and $x$ .

$\frac{x^{2} + 9 x + 18 + 2 x - 1 - x^{2}}{(x + 3) (x - 1)}$

Subtract $x^{2}$ from $x^{2}$ .

$\frac{9 x + 18 + 2 x - 1 + 0}{(x + 3) (x - 1)}$

Add $9 x + 18 + 2 x - 1$ and $0$ .

$\frac{9 x + 18 + 2 x - 1}{(x + 3) (x - 1)}$

Add $9 x$ and $2 x$ .

$\frac{11 x + 18 - 1}{(x + 3) (x - 1)}$

Subtract $1$ from $18$ .

$\frac{11 x + 17}{(x + 3) (x - 1)}$

Simplify (x+6)/(x-1)+(1-x)/(x+3)

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