Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

To divide by a fraction, multiply by its reciprocal.

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

Step 1.2

Factor $x$ out of $x^{2} + 3 x$ .

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Step 1.2.1

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

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

Step 1.2.2

Factor $x$ out of $3 x$ .

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

Step 1.2.3

Factor $x$ out of $x \cdot x + x \cdot 3$ .

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

Step 1.3

Simplify the denominator.

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Step 1.3.1

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

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

Step 1.3.2

Rewrite $1$ as $1^{2}$ .

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

Step 1.3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 1$ .

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

Step 1.4

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

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Step 1.4.1

Factor $2$ out of $4 x$ .

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

Step 1.4.2

Factor $2$ out of $2$ .

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

Step 1.4.3

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

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

Step 1.5

Cancel the common factor of $2 x + 1$ .

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Step 1.5.1

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

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

Step 1.5.2

Cancel the common factor.

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

Step 1.5.3

Rewrite the expression.

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

Step 1.6

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

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

Step 1.7

Cancel the common factor of $x + 3$ and $3 + x$ .

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Step 1.7.1

Reorder terms.

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

Step 1.7.2

Cancel the common factor.

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

Step 1.7.3

Rewrite the expression.

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

Step 1.8

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

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

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Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

Apply the distributive property.

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

Step 6.1.3

Apply the distributive property.

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

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 $2 x$ by $1$ .

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

Step 6.2.1.2

Multiply $- 1$ by $1$ .

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

Step 6.2.1.3

Rewrite using the commutative property of multiplication.

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

Step 6.2.1.4

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

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Step 6.2.1.4.1

Move $x$ .

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

Step 6.2.1.4.2

Multiply $x$ by $x$ .

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

Step 6.2.1.5

Multiply $- 2$ by $2$ .

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

Step 6.2.1.6

Multiply $- 1$ by $- 2$ .

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

Step 6.2.2

Add $2 x$ and $2 x$ .

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

Step 6.3

Apply the distributive property.

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

Step 6.4

Rewrite using the commutative property of multiplication.

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

Step 6.5

Multiply $2$ by $1$ .

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

Step 6.6

Simplify each term.

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Step 6.6.1

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

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Step 6.6.1.1

Move $x$ .

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

Step 6.6.1.2

Multiply $x$ by $x$ .

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

Step 6.6.2

Multiply $2$ by $2$ .

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

Step 6.7

Add $4 x$ and $2 x$ .

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

Step 6.8

Add $- 4 x^{2}$ and $4 x^{2}$ .

$\frac{- 1 + 0 + 6 x}{(2 x + 1) (2 x - 1)}$

Step 6.9

Add $- 1$ and $0$ .

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

Step 7

Simplify with factoring out.

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Step 7.1

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

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

Step 7.2

Factor $- 1$ out of $6 x$ .

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

Step 7.3

Factor $- 1$ out of $- 1 (1) - (- 6 x)$ .

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

Step 7.4

Move the negative in front of the fraction.

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