Algebra Examples

\frac{7}{x + 1} + \frac{2}{x + 6}

$\frac{7}{x + 1} + \frac{2}{x + 6}$

Step 1

To write

\frac{7}{x + 1}

$\frac{7}{x + 1}$ as a fraction with a common denominator, multiply by

\frac{x + 6}{x + 6}

$\frac{x + 6}{x + 6}$ .

\frac{7}{x + 1} \cdot \frac{x + 6}{x + 6} + \frac{2}{x + 6}

$\frac{7}{x + 1} \cdot \frac{x + 6}{x + 6} + \frac{2}{x + 6}$

Step 2

To write

\frac{2}{x + 6}

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

\frac{x + 1}{x + 1}

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

\frac{7}{x + 1} \cdot \frac{x + 6}{x + 6} + \frac{2}{x + 6} \cdot \frac{x + 1}{x + 1}

$\frac{7}{x + 1} \cdot \frac{x + 6}{x + 6} + \frac{2}{x + 6} \cdot \frac{x + 1}{x + 1}$

Step 3

Write each expression with a common denominator of

(x + 1) (x + 6)

$(x + 1) (x + 6)$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 3.1

Multiply

\frac{7}{x + 1}

$\frac{7}{x + 1}$ by

\frac{x + 6}{x + 6}

$\frac{x + 6}{x + 6}$ .

\frac{7 (x + 6)}{(x + 1) (x + 6)} + \frac{2}{x + 6} \cdot \frac{x + 1}{x + 1}

$\frac{7 (x + 6)}{(x + 1) (x + 6)} + \frac{2}{x + 6} \cdot \frac{x + 1}{x + 1}$

Step 3.2

Multiply

\frac{2}{x + 6}

$\frac{2}{x + 6}$ by

\frac{x + 1}{x + 1}

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

\frac{7 (x + 6)}{(x + 1) (x + 6)} + \frac{2 (x + 1)}{(x + 6) (x + 1)}

$\frac{7 (x + 6)}{(x + 1) (x + 6)} + \frac{2 (x + 1)}{(x + 6) (x + 1)}$

Step 3.3

Reorder the factors of

(x + 6) (x + 1)

$(x + 6) (x + 1)$ .

\frac{7 (x + 6)}{(x + 1) (x + 6)} + \frac{2 (x + 1)}{(x + 1) (x + 6)}

$\frac{7 (x + 6)}{(x + 1) (x + 6)} + \frac{2 (x + 1)}{(x + 1) (x + 6)}$

\frac{7 (x + 6)}{(x + 1) (x + 6)} + \frac{2 (x + 1)}{(x + 1) (x + 6)}

$\frac{7 (x + 6)}{(x + 1) (x + 6)} + \frac{2 (x + 1)}{(x + 1) (x + 6)}$

Step 4

Combine the numerators over the common denominator.

\frac{7 (x + 6) + 2 (x + 1)}{(x + 1) (x + 6)}

$\frac{7 (x + 6) + 2 (x + 1)}{(x + 1) (x + 6)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

\frac{7 x + 7 \cdot 6 + 2 (x + 1)}{(x + 1) (x + 6)}

$\frac{7 x + 7 \cdot 6 + 2 (x + 1)}{(x + 1) (x + 6)}$

Step 5.2

Multiply

7

$7$ by

6

$6$ .

\frac{7 x + 42 + 2 (x + 1)}{(x + 1) (x + 6)}

$\frac{7 x + 42 + 2 (x + 1)}{(x + 1) (x + 6)}$

Step 5.3

Apply the distributive property.

\frac{7 x + 42 + 2 x + 2 \cdot 1}{(x + 1) (x + 6)}

$\frac{7 x + 42 + 2 x + 2 \cdot 1}{(x + 1) (x + 6)}$

Step 5.4

Multiply

2

$2$ by

1

$1$ .

\frac{7 x + 42 + 2 x + 2}{(x + 1) (x + 6)}

$\frac{7 x + 42 + 2 x + 2}{(x + 1) (x + 6)}$

Step 5.5

Add

7 x

$7 x$ and

2 x

$2 x$ .

\frac{9 x + 42 + 2}{(x + 1) (x + 6)}

$\frac{9 x + 42 + 2}{(x + 1) (x + 6)}$

Step 5.6

Add

42

$42$ and

2

$2$ .

\frac{9 x + 44}{(x + 1) (x + 6)}

$\frac{9 x + 44}{(x + 1) (x + 6)}$

\frac{9 x + 44}{(x + 1) (x + 6)}

$\frac{9 x + 44}{(x + 1) (x + 6)}$

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