Simplify (x+5)/(3x+4)+(x+4)/(x+3)

$\frac{x + 5}{3 x + 4} + \frac{x + 4}{x + 3}$

Step 1

To write $\frac{x + 5}{3 x + 4}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$\frac{x + 5}{3 x + 4} \cdot \frac{x + 3}{x + 3} + \frac{x + 4}{x + 3}$

Step 2

To write $\frac{x + 4}{x + 3}$ as a fraction with a common denominator, multiply by $\frac{3 x + 4}{3 x + 4}$ .

$\frac{x + 5}{3 x + 4} \cdot \frac{x + 3}{x + 3} + \frac{x + 4}{x + 3} \cdot \frac{3 x + 4}{3 x + 4}$

Step 3

Write each expression with a common denominator of $(3 x + 4) (x + 3)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x + 5}{3 x + 4}$ by $\frac{x + 3}{x + 3}$ .

$\frac{(x + 5) (x + 3)}{(3 x + 4) (x + 3)} + \frac{x + 4}{x + 3} \cdot \frac{3 x + 4}{3 x + 4}$

Multiply $\frac{x + 4}{x + 3}$ by $\frac{3 x + 4}{3 x + 4}$ .

$\frac{(x + 5) (x + 3)}{(3 x + 4) (x + 3)} + \frac{(x + 4) (3 x + 4)}{(x + 3) (3 x + 4)}$

Reorder the factors of $(x + 3) (3 x + 4)$ .

$\frac{(x + 5) (x + 3)}{(3 x + 4) (x + 3)} + \frac{(x + 4) (3 x + 4)}{(3 x + 4) (x + 3)}$

Step 4

Combine the numerators over the common denominator.

$\frac{(x + 5) (x + 3) + (x + 4) (3 x + 4)}{(3 x + 4) (x + 3)}$

Step 5

Simplify the numerator.

Expand $(x + 5) (x + 3)$ using the FOIL Method.

Apply the distributive property.

$\frac{x (x + 3) + 5 (x + 3) + (x + 4) (3 x + 4)}{(3 x + 4) (x + 3)}$

Apply the distributive property.

$\frac{x \cdot x + x \cdot 3 + 5 (x + 3) + (x + 4) (3 x + 4)}{(3 x + 4) (x + 3)}$

Apply the distributive property.

$\frac{x \cdot x + x \cdot 3 + 5 x + 5 \cdot 3 + (x + 4) (3 x + 4)}{(3 x + 4) (x + 3)}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 3 + 5 x + 5 \cdot 3 + (x + 4) (3 x + 4)}{(3 x + 4) (x + 3)}$

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

$\frac{x^{2} + 3 \cdot x + 5 x + 5 \cdot 3 + (x + 4) (3 x + 4)}{(3 x + 4) (x + 3)}$

Multiply $5$ by $3$ .

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

Add $3 x$ and $5 x$ .

$\frac{x^{2} + 8 x + 15 + (x + 4) (3 x + 4)}{(3 x + 4) (x + 3)}$

Expand $(x + 4) (3 x + 4)$ using the FOIL Method.

Apply the distributive property.

$\frac{x^{2} + 8 x + 15 + x (3 x + 4) + 4 (3 x + 4)}{(3 x + 4) (x + 3)}$

Apply the distributive property.

$\frac{x^{2} + 8 x + 15 + x (3 x) + x \cdot 4 + 4 (3 x + 4)}{(3 x + 4) (x + 3)}$

Apply the distributive property.

$\frac{x^{2} + 8 x + 15 + x (3 x) + x \cdot 4 + 4 (3 x) + 4 \cdot 4}{(3 x + 4) (x + 3)}$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$\frac{x^{2} + 8 x + 15 + 3 x \cdot x + x \cdot 4 + 4 (3 x) + 4 \cdot 4}{(3 x + 4) (x + 3)}$

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

Move $x$ .

$\frac{x^{2} + 8 x + 15 + 3 (x \cdot x) + x \cdot 4 + 4 (3 x) + 4 \cdot 4}{(3 x + 4) (x + 3)}$

Multiply $x$ by $x$ .

$\frac{x^{2} + 8 x + 15 + 3 x^{2} + x \cdot 4 + 4 (3 x) + 4 \cdot 4}{(3 x + 4) (x + 3)}$

Move $4$ to the left of $x$ .

$\frac{x^{2} + 8 x + 15 + 3 x^{2} + 4 \cdot x + 4 (3 x) + 4 \cdot 4}{(3 x + 4) (x + 3)}$

Multiply $3$ by $4$ .

$\frac{x^{2} + 8 x + 15 + 3 x^{2} + 4 x + 12 x + 4 \cdot 4}{(3 x + 4) (x + 3)}$

Multiply $4$ by $4$ .

$\frac{x^{2} + 8 x + 15 + 3 x^{2} + 4 x + 12 x + 16}{(3 x + 4) (x + 3)}$

Add $4 x$ and $12 x$ .

$\frac{x^{2} + 8 x + 15 + 3 x^{2} + 16 x + 16}{(3 x + 4) (x + 3)}$

Add $x^{2}$ and $3 x^{2}$ .

$\frac{4 x^{2} + 8 x + 15 + 16 x + 16}{(3 x + 4) (x + 3)}$

Add $8 x$ and $16 x$ .

$\frac{4 x^{2} + 24 x + 15 + 16}{(3 x + 4) (x + 3)}$

Add $15$ and $16$ .

$\frac{4 x^{2} + 24 x + 31}{(3 x + 4) (x + 3)}$

Simplify (x+5)/(3x+4)+(x+4)/(x+3)

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