Simplify (x+2)/(x+4)-(x-1)/(x+6)

$\frac{x + 2}{x + 4} - \frac{x - 1}{x + 6}$

Step 1

To write $\frac{x + 2}{x + 4}$ as a fraction with a common denominator, multiply by $\frac{x + 6}{x + 6}$ .

$\frac{x + 2}{x + 4} \cdot \frac{x + 6}{x + 6} - \frac{x - 1}{x + 6}$

Step 2

To write $- \frac{x - 1}{x + 6}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$\frac{x + 2}{x + 4} \cdot \frac{x + 6}{x + 6} - \frac{x - 1}{x + 6} \cdot \frac{x + 4}{x + 4}$

Step 3

Write each expression with a common denominator of $(x + 4) (x + 6)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x + 2}{x + 4}$ by $\frac{x + 6}{x + 6}$ .

$\frac{(x + 2) (x + 6)}{(x + 4) (x + 6)} - \frac{x - 1}{x + 6} \cdot \frac{x + 4}{x + 4}$

Multiply $\frac{x - 1}{x + 6}$ by $\frac{x + 4}{x + 4}$ .

$\frac{(x + 2) (x + 6)}{(x + 4) (x + 6)} - \frac{(x - 1) (x + 4)}{(x + 6) (x + 4)}$

Reorder the factors of $(x + 6) (x + 4)$ .

$\frac{(x + 2) (x + 6)}{(x + 4) (x + 6)} - \frac{(x - 1) (x + 4)}{(x + 4) (x + 6)}$

Step 4

Combine the numerators over the common denominator.

$\frac{(x + 2) (x + 6) - (x - 1) (x + 4)}{(x + 4) (x + 6)}$

Step 5

Simplify the numerator.

Expand $(x + 2) (x + 6)$ using the FOIL Method.

Apply the distributive property.

$\frac{x (x + 6) + 2 (x + 6) - (x - 1) (x + 4)}{(x + 4) (x + 6)}$

Apply the distributive property.

$\frac{x \cdot x + x \cdot 6 + 2 (x + 6) - (x - 1) (x + 4)}{(x + 4) (x + 6)}$

Apply the distributive property.

$\frac{x \cdot x + x \cdot 6 + 2 x + 2 \cdot 6 - (x - 1) (x + 4)}{(x + 4) (x + 6)}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 6 + 2 x + 2 \cdot 6 - (x - 1) (x + 4)}{(x + 4) (x + 6)}$

Move $6$ to the left of $x$ .

$\frac{x^{2} + 6 \cdot x + 2 x + 2 \cdot 6 - (x - 1) (x + 4)}{(x + 4) (x + 6)}$

Multiply $2$ by $6$ .

$\frac{x^{2} + 6 x + 2 x + 12 - (x - 1) (x + 4)}{(x + 4) (x + 6)}$

Add $6 x$ and $2 x$ .

$\frac{x^{2} + 8 x + 12 - (x - 1) (x + 4)}{(x + 4) (x + 6)}$

Apply the distributive property.

$\frac{x^{2} + 8 x + 12 + (- x - - 1) (x + 4)}{(x + 4) (x + 6)}$

Multiply $- 1$ by $- 1$ .

$\frac{x^{2} + 8 x + 12 + (- x + 1) (x + 4)}{(x + 4) (x + 6)}$

Expand $(- x + 1) (x + 4)$ using the FOIL Method.

Apply the distributive property.

$\frac{x^{2} + 8 x + 12 - x (x + 4) + 1 (x + 4)}{(x + 4) (x + 6)}$

Apply the distributive property.

$\frac{x^{2} + 8 x + 12 - x \cdot x - x \cdot 4 + 1 (x + 4)}{(x + 4) (x + 6)}$

Apply the distributive property.

$\frac{x^{2} + 8 x + 12 - x \cdot x - x \cdot 4 + 1 x + 1 \cdot 4}{(x + 4) (x + 6)}$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$\frac{x^{2} + 8 x + 12 - (x \cdot x) - x \cdot 4 + 1 x + 1 \cdot 4}{(x + 4) (x + 6)}$

Multiply $x$ by $x$ .

$\frac{x^{2} + 8 x + 12 - x^{2} - x \cdot 4 + 1 x + 1 \cdot 4}{(x + 4) (x + 6)}$

Multiply $4$ by $- 1$ .

$\frac{x^{2} + 8 x + 12 - x^{2} - 4 x + 1 x + 1 \cdot 4}{(x + 4) (x + 6)}$

Multiply $x$ by $1$ .

$\frac{x^{2} + 8 x + 12 - x^{2} - 4 x + x + 1 \cdot 4}{(x + 4) (x + 6)}$

Multiply $4$ by $1$ .

$\frac{x^{2} + 8 x + 12 - x^{2} - 4 x + x + 4}{(x + 4) (x + 6)}$

Add $- 4 x$ and $x$ .

$\frac{x^{2} + 8 x + 12 - x^{2} - 3 x + 4}{(x + 4) (x + 6)}$

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

$\frac{8 x + 12 + 0 - 3 x + 4}{(x + 4) (x + 6)}$

Add $8 x$ and $0$ .

$\frac{8 x + 12 - 3 x + 4}{(x + 4) (x + 6)}$

Subtract $3 x$ from $8 x$ .

$\frac{5 x + 12 + 4}{(x + 4) (x + 6)}$

Add $12$ and $4$ .

$\frac{5 x + 16}{(x + 4) (x + 6)}$

Simplify (x+2)/(x+4)-(x-1)/(x+6)

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