Algebra Examples

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

Step 1

To write $\frac{x + 1}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

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

Step 2

To write $- \frac{x + 3}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

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

Step 3

Write each expression with a common denominator of $(x - 4) (x + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{x + 1}{x - 4}$ by $\frac{x + 2}{x + 2}$ .

$\frac{(x + 1) (x + 2)}{(x - 4) (x + 2)} - \frac{x + 3}{x + 2} \cdot \frac{x - 4}{x - 4}$

Step 3.2

Multiply $\frac{x + 3}{x + 2}$ by $\frac{x - 4}{x - 4}$ .

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

Step 3.3

Reorder the factors of $(x - 4) (x + 2)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Expand $(x + 1) (x + 2)$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot 2 + 1 (x + 2) - (x + 3) (x - 4)}{(x + 2) (x - 4)}$

Step 5.1.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot 2 + 1 x + 1 \cdot 2 - (x + 3) (x - 4)}{(x + 2) (x - 4)}$

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 $x$ .

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

Step 5.2.1.2

Move $2$ to the left of $x$ .

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

Step 5.2.1.3

Multiply $x$ by $1$ .

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

Step 5.2.1.4

Multiply $2$ by $1$ .

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

Step 5.2.2

Add $2 x$ and $x$ .

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

Step 5.3

Apply the distributive property.

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

Step 5.4

Multiply $- 1$ by $3$ .

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

Step 5.5

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

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Step 5.5.1

Apply the distributive property.

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

Step 5.5.2

Apply the distributive property.

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

Step 5.5.3

Apply the distributive property.

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

Step 5.6

Simplify and combine like terms.

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Step 5.6.1

Simplify each term.

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Step 5.6.1.1

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

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Step 5.6.1.1.1

Move $x$ .

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

Step 5.6.1.1.2

Multiply $x$ by $x$ .

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

Step 5.6.1.2

Multiply $- 4$ by $- 1$ .

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

Step 5.6.1.3

Multiply $- 3$ by $- 4$ .

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

Step 5.6.2

Subtract $3 x$ from $4 x$ .

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

Step 5.7

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

$\frac{3 x + 2 + 0 + x + 12}{(x + 2) (x - 4)}$

Step 5.8

Add $3 x$ and $0$ .

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

Step 5.9

Add $3 x$ and $x$ .

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

Step 5.10

Add $2$ and $12$ .

$\frac{4 x + 14}{(x + 2) (x - 4)}$

Step 5.11

Factor $2$ out of $4 x + 14$ .

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Step 5.11.1

Factor $2$ out of $4 x$ .

$\frac{2 (2 x) + 14}{(x + 2) (x - 4)}$

Step 5.11.2

Factor $2$ out of $14$ .

$\frac{2 (2 x) + 2 (7)}{(x + 2) (x - 4)}$

Step 5.11.3

Factor $2$ out of $2 (2 x) + 2 (7)$ .

$\frac{2 (2 x + 7)}{(x + 2) (x - 4)}$