Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $x$ by $x$ .

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

Step 5.2.1.2

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

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

Step 5.2.1.3

Multiply $- 5$ by $3$ .

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

Step 5.2.2

Subtract $5 x$ from $3 x$ .

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

Step 5.3

Apply the distributive property.

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

Step 5.4

Multiply $- 3$ by $4$ .

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

Step 5.5

Subtract $3 x$ from $- 2 x$ .

$\frac{x^{2} - 5 x - 15 - 12}{(x + 3) (x + 4)}$

Step 5.6

Subtract $12$ from $- 15$ .

$\frac{x^{2} - 5 x - 27}{(x + 3) (x + 4)}$