Algebra Examples

$\frac{x + 1}{x + 2} + \frac{1}{\frac{x + 1}{x + 2}}$

Step 1

Simplify each term.

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Step 1.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x + 1}{x + 2} + 1 \frac{x + 2}{x + 1}$

Step 1.2

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

$\frac{x + 1}{x + 2} + \frac{x + 2}{x + 1}$

Step 2

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

$\frac{x + 1}{x + 2} \cdot \frac{x + 1}{x + 1} + \frac{x + 2}{x + 1}$

Step 3

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

$\frac{x + 1}{x + 2} \cdot \frac{x + 1}{x + 1} + \frac{x + 2}{x + 1} \cdot \frac{x + 2}{x + 2}$

Step 4

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

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Step 4.1

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

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

Step 4.2

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

$\frac{(x + 1) (x + 1)}{(x + 2) (x + 1)} + \frac{(x + 2) (x + 2)}{(x + 1) (x + 2)}$

Step 4.3

Reorder the factors of $(x + 2) (x + 1)$ .

$\frac{(x + 1) (x + 1)}{(x + 1) (x + 2)} + \frac{(x + 2) (x + 2)}{(x + 1) (x + 2)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(x + 1) (x + 1) + (x + 2) (x + 2)}{(x + 1) (x + 2)}$

Step 6

Simplify the numerator.

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Step 6.1

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

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Step 6.1.1

Apply the distributive property.

$\frac{x (x + 1) + 1 (x + 1) + (x + 2) (x + 2)}{(x + 1) (x + 2)}$

Step 6.1.2

Apply the distributive property.

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

Step 6.1.3

Apply the distributive property.

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

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $x$ by $x$ .

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

Step 6.2.1.2

Multiply $x$ by $1$ .

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

Step 6.2.1.3

Multiply $x$ by $1$ .

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

Step 6.2.1.4

Multiply $1$ by $1$ .

$\frac{x^{2} + x + x + 1 + (x + 2) (x + 2)}{(x + 1) (x + 2)}$

Step 6.2.2

Add $x$ and $x$ .

$\frac{x^{2} + 2 x + 1 + (x + 2) (x + 2)}{(x + 1) (x + 2)}$

Step 6.3

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

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Step 6.3.1

Apply the distributive property.

$\frac{x^{2} + 2 x + 1 + x (x + 2) + 2 (x + 2)}{(x + 1) (x + 2)}$

Step 6.3.2

Apply the distributive property.

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

Step 6.3.3

Apply the distributive property.

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

Step 6.4

Simplify and combine like terms.

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Step 6.4.1

Simplify each term.

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Step 6.4.1.1

Multiply $x$ by $x$ .

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

Step 6.4.1.2

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

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

Step 6.4.1.3

Multiply $2$ by $2$ .

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

Step 6.4.2

Add $2 x$ and $2 x$ .

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

Step 6.5

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

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

Step 6.6

Add $2 x$ and $4 x$ .

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

Step 6.7

Add $1$ and $4$ .

$\frac{2 x^{2} + 6 x + 5}{(x + 1) (x + 2)}$