Algebra Examples

$\frac{1 + \frac{1}{10 - b}}{1 - \frac{1}{10 + b}}$

Step 1

Multiply the numerator and denominator of the fraction by $(10 - b) (10 + b)$ .

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Step 1.1

Multiply $\frac{1 + \frac{1}{10 - b}}{1 - \frac{1}{10 + b}}$ by $\frac{(10 - b) (10 + b)}{(10 - b) (10 + b)}$ .

$\frac{(10 - b) (10 + b)}{(10 - b) (10 + b)} \cdot \frac{1 + \frac{1}{10 - b}}{1 - \frac{1}{10 + b}}$

Step 1.2

Combine.

$\frac{(10 - b) (10 + b) (1 + \frac{1}{10 - b})}{(10 - b) (10 + b) (1 - \frac{1}{10 + b})}$

Step 2

Apply the distributive property.

$\frac{(10 - b) (10 + b) \cdot 1 + (10 - b) (10 + b) \frac{1}{10 - b}}{(10 - b) (10 + b) \cdot 1 + (10 - b) (10 + b) (- \frac{1}{10 + b})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $10 - b$ .

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Step 3.1.1

Cancel the common factor.

$\frac{(10 - b) (10 + b) \cdot 1 + (10 - b) (10 + b) \frac{1}{10 - b}}{(10 - b) (10 + b) \cdot 1 + (10 - b) (10 + b) (- \frac{1}{10 + b})}$

Step 3.1.2

Rewrite the expression.

$\frac{(10 - b) (10 + b) \cdot 1 + 10 + b}{(10 - b) (10 + b) \cdot 1 + (10 - b) (10 + b) (- \frac{1}{10 + b})}$

Step 3.2

Cancel the common factor of $10 + b$ .

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Step 3.2.1

Move the leading negative in $- \frac{1}{10 + b}$ into the numerator.

$\frac{(10 - b) (10 + b) \cdot 1 + 10 + b}{(10 - b) (10 + b) \cdot 1 + (10 - b) (10 + b) \frac{- 1}{10 + b}}$

Step 3.2.2

Factor $10 + b$ out of $(10 - b) (10 + b)$ .

$\frac{(10 - b) (10 + b) \cdot 1 + 10 + b}{(10 - b) (10 + b) \cdot 1 + (10 + b) (10 - b) \frac{- 1}{10 + b}}$

Step 3.2.3

Cancel the common factor.

$\frac{(10 - b) (10 + b) \cdot 1 + 10 + b}{(10 - b) (10 + b) \cdot 1 + (10 + b) (10 - b) \frac{- 1}{10 + b}}$

Step 3.2.4

Rewrite the expression.

$\frac{(10 - b) (10 + b) \cdot 1 + 10 + b}{(10 - b) (10 + b) \cdot 1 + (10 - b) \cdot - 1}$

Step 4

Simplify the numerator.

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Step 4.1

Expand $(10 - b) (10 + b)$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

$\frac{(10 (10 + b) - b (10 + b)) \cdot 1 + 10 + b}{(10 - b) (10 + b) \cdot 1 + (10 - b) \cdot - 1}$

Step 4.1.2

Apply the distributive property.

$\frac{(10 \cdot 10 + 10 b - b (10 + b)) \cdot 1 + 10 + b}{(10 - b) (10 + b) \cdot 1 + (10 - b) \cdot - 1}$

Step 4.1.3

Apply the distributive property.

$\frac{(10 \cdot 10 + 10 b - b \cdot 10 - b \cdot b) \cdot 1 + 10 + b}{(10 - b) (10 + b) \cdot 1 + (10 - b) \cdot - 1}$

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $10$ by $10$ .

$\frac{(100 + 10 b - b \cdot 10 - b \cdot b) \cdot 1 + 10 + b}{(10 - b) (10 + b) \cdot 1 + (10 - b) \cdot - 1}$

Step 4.2.1.2

Multiply $10$ by $- 1$ .

$\frac{(100 + 10 b - 10 b - b \cdot b) \cdot 1 + 10 + b}{(10 - b) (10 + b) \cdot 1 + (10 - b) \cdot - 1}$

Step 4.2.1.3

Multiply $b$ by $b$ by adding the exponents.

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Step 4.2.1.3.1

Move $b$ .

$\frac{(100 + 10 b - 10 b - (b \cdot b)) \cdot 1 + 10 + b}{(10 - b) (10 + b) \cdot 1 + (10 - b) \cdot - 1}$

Step 4.2.1.3.2

Multiply $b$ by $b$ .

$\frac{(100 + 10 b - 10 b - b^{2}) \cdot 1 + 10 + b}{(10 - b) (10 + b) \cdot 1 + (10 - b) \cdot - 1}$

Step 4.2.2

Subtract $10 b$ from $10 b$ .

$\frac{(100 + 0 - b^{2}) \cdot 1 + 10 + b}{(10 - b) (10 + b) \cdot 1 + (10 - b) \cdot - 1}$

Step 4.2.3

Add $100$ and $0$ .

$\frac{(100 - b^{2}) \cdot 1 + 10 + b}{(10 - b) (10 + b) \cdot 1 + (10 - b) \cdot - 1}$

Step 4.3

Multiply $100 - b^{2}$ by $1$ .

$\frac{100 - b^{2} + 10 + b}{(10 - b) (10 + b) \cdot 1 + (10 - b) \cdot - 1}$

Step 4.4

Add $100$ and $10$ .

$\frac{- b^{2} + 110 + b}{(10 - b) (10 + b) \cdot 1 + (10 - b) \cdot - 1}$

Step 4.5

Reorder terms.

$\frac{- b^{2} + b + 110}{(10 - b) (10 + b) \cdot 1 + (10 - b) \cdot - 1}$

Step 4.6

Factor by grouping.

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Step 4.6.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 110 = - 110$ and whose sum is $b = 1$ .

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Step 4.6.1.1

Multiply by $1$ .

$\frac{- b^{2} + 1 b + 110}{(10 - b) (10 + b) \cdot 1 + (10 - b) \cdot - 1}$

Step 4.6.1.2

Rewrite $1$ as $- 10$ plus $11$

$\frac{- b^{2} + (- 10 + 11) b + 110}{(10 - b) (10 + b) \cdot 1 + (10 - b) \cdot - 1}$

Step 4.6.1.3

Apply the distributive property.

$\frac{- b^{2} - 10 b + 11 b + 110}{(10 - b) (10 + b) \cdot 1 + (10 - b) \cdot - 1}$

Step 4.6.2

Factor out the greatest common factor from each group.

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Step 4.6.2.1

Group the first two terms and the last two terms.

$\frac{(- b^{2} - 10 b) + 11 b + 110}{(10 - b) (10 + b) \cdot 1 + (10 - b) \cdot - 1}$

Step 4.6.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{b (- b - 10) - 11 (- b - 10)}{(10 - b) (10 + b) \cdot 1 + (10 - b) \cdot - 1}$

Step 4.6.3

Factor the polynomial by factoring out the greatest common factor, $- b - 10$ .

$\frac{(- b - 10) (b - 11)}{(10 - b) (10 + b) \cdot 1 + (10 - b) \cdot - 1}$

Step 5

Simplify the denominator.

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Step 5.1

Factor $10 - b$ out of $(10 - b) (10 + b) \cdot 1 + (10 - b) \cdot - 1$ .

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Step 5.1.1

Factor $10 - b$ out of $(10 - b) (10 + b) \cdot 1$ .

$\frac{(- b - 10) (b - 11)}{(10 - b) ((10 + b) \cdot 1) + (10 - b) \cdot - 1}$

Step 5.1.2

Factor $10 - b$ out of $(10 - b) \cdot - 1$ .

$\frac{(- b - 10) (b - 11)}{(10 - b) ((10 + b) \cdot 1) + (10 - b) (- 1)}$

Step 5.1.3

Factor $10 - b$ out of $(10 - b) ((10 + b) \cdot 1) + (10 - b) (- 1)$ .

$\frac{(- b - 10) (b - 11)}{(10 - b) ((10 + b) \cdot 1 - 1)}$

Step 5.2

Multiply $10 + b$ by $1$ .

$\frac{(- b - 10) (b - 11)}{(10 - b) (10 + b - 1)}$

Step 5.3

Subtract $1$ from $10$ .

$\frac{(- b - 10) (b - 11)}{(10 - b) (b + 9)}$

Step 6

Simplify with factoring out.

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Step 6.1

Factor $- 1$ out of $- b$ .

$\frac{(- (b) - 10) (b - 11)}{(10 - b) (b + 9)}$

Step 6.2

Rewrite $- 10$ as $- 1 (10)$ .

$\frac{(- (b) - 1 (10)) (b - 11)}{(10 - b) (b + 9)}$

Step 6.3

Factor $- 1$ out of $- (b) - 1 (10)$ .

$\frac{- (b + 10) (b - 11)}{(10 - b) (b + 9)}$

Step 6.4

Simplify the expression.

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Step 6.4.1

Rewrite $- (b + 10)$ as $- 1 (b + 10)$ .

$\frac{- 1 (b + 10) (b - 11)}{(10 - b) (b + 9)}$

Step 6.4.2

Move the negative in front of the fraction.

$- \frac{(b + 10) (b - 11)}{(10 - b) (b + 9)}$