Precalculus Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $(x + 2) (x + 1) x$ .

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Step 1.1

Multiply $\frac{\frac{x - 2}{x + 2} + \frac{x - 1}{x + 1}}{\frac{x}{x + 1} - \frac{2 x - 3}{x}}$ by $\frac{(x + 2) (x + 1) x}{(x + 2) (x + 1) x}$ .

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x + 2$ .

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Step 3.1.1

Factor $x + 2$ out of $(x + 2) (x + 1) x$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x + 1$ .

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Step 3.2.1

Factor $x + 1$ out of $(x + 2) (x + 1) x$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $x + 1$ .

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Step 3.3.1

Factor $x + 1$ out of $(x + 2) (x + 1) x$ .

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

Raise $x$ to the power of $1$ .

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

Step 3.5

Raise $x$ to the power of $1$ .

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

Step 3.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.7

Add $1$ and $1$ .

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

Step 3.8

Cancel the common factor of $x$ .

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Step 3.8.1

Move the leading negative in $- \frac{2 x - 3}{x}$ into the numerator.

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

Step 3.8.2

Factor $x$ out of $(x + 2) (x + 1) x$ .

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

Step 3.8.3

Cancel the common factor.

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

Step 3.8.4

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Factor $x$ out of $(x + 1) x (x - 2) + (x + 2) x (x - 1)$ .

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Step 4.1.1

Factor $x$ out of $(x + 1) x (x - 2)$ .

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

Step 4.1.2

Factor $x$ out of $(x + 2) x (x - 1)$ .

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

Step 4.1.3

Factor $x$ out of $x ((x + 1) (x - 2)) + x ((x + 2) (x - 1))$ .

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

Step 4.2

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

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Step 4.2.1

Apply the distributive property.

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

Step 4.2.2

Apply the distributive property.

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

Step 4.2.3

Apply the distributive property.

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

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply $x$ by $x$ .

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

Step 4.3.1.2

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

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

Step 4.3.1.3

Multiply $x$ by $1$ .

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

Step 4.3.1.4

Multiply $- 2$ by $1$ .

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

Step 4.3.2

Add $- 2 x$ and $x$ .

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

Step 4.4

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

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Step 4.4.1

Apply the distributive property.

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

Step 4.4.2

Apply the distributive property.

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

Step 4.4.3

Apply the distributive property.

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

Step 4.5

Simplify and combine like terms.

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Step 4.5.1

Simplify each term.

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Step 4.5.1.1

Multiply $x$ by $x$ .

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

Step 4.5.1.2

Move $- 1$ to the left of $x$ .

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

Step 4.5.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 4.5.1.4

Multiply $2$ by $- 1$ .

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

Step 4.5.2

Add $- x$ and $2 x$ .

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

Step 4.6

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

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

Step 4.7

Add $- x$ and $x$ .

$\frac{x (2 x^{2} + 0 - 2 - 2)}{(x + 2) x^{2} + (x + 2) (x + 1) (- (2 x - 3))}$

Step 4.8

Add $2 x^{2}$ and $0$ .

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

Step 4.9

Subtract $2$ from $- 2$ .

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

Step 4.10

Factor $2$ out of $2 x^{2} - 4$ .

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Step 4.10.1

Factor $2$ out of $2 x^{2}$ .

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

Step 4.10.2

Factor $2$ out of $- 4$ .

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

Step 4.10.3

Factor $2$ out of $2 x^{2} + 2 \cdot - 2$ .

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

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

Step 5

Simplify the denominator.

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Step 5.1

Factor $x + 2$ out of $(x + 2) x^{2} + (x + 2) (x + 1) (- (2 x - 3))$ .

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Step 5.1.1

Factor $x + 2$ out of $(x + 2) x^{2}$ .

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

Step 5.1.2

Factor $x + 2$ out of $(x + 2) (x + 1) (- (2 x - 3))$ .

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

Step 5.1.3

Factor $x + 2$ out of $(x + 2) (x^{2}) + (x + 2) ((x + 1) (- (2 x - 3)))$ .

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Multiply $2$ by $- 1$ .

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

Step 5.4

Multiply $- 1$ by $- 3$ .

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

Step 5.5

Expand $(x + 1) (- 2 x + 3)$ using the FOIL Method.

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Step 5.5.1

Apply the distributive property.

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

Step 5.5.2

Apply the distributive property.

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

Step 5.5.3

Apply the distributive property.

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

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

Rewrite using the commutative property of multiplication.

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

Step 5.6.1.2

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

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Step 5.6.1.2.1

Move $x$ .

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

Step 5.6.1.2.2

Multiply $x$ by $x$ .

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

Step 5.6.1.3

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

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

Step 5.6.1.4

Multiply $- 2 x$ by $1$ .

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

Step 5.6.1.5

Multiply $3$ by $1$ .

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

Step 5.6.2

Subtract $2 x$ from $3 x$ .

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

Step 5.7

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

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

Step 6

Simplify with factoring out.

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Step 6.1

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

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

Step 6.2

Factor $- 1$ out of $- x^{2}$ .

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

Step 6.3

Factor $- 1$ out of $x$ .

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

Step 6.4

Factor $- 1$ out of $- (x^{2}) - 1 (- x)$ .

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

Step 6.5

Rewrite $3$ as $- 1 (- 3)$ .

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

Step 6.6

Factor $- 1$ out of $- (x^{2} - x) - 1 (- 3)$ .

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

Step 6.7

Rewrite negatives.

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Step 6.7.1

Rewrite $- (x^{2} - x - 3)$ as $- 1 (x^{2} - x - 3)$ .

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

Step 6.7.2

Move the negative in front of the fraction.

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