Algebra Examples

$\frac{4 x + 1}{6 x} \div \frac{x}{3 x - 1}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{4 x + 1}{6 x} \cdot \frac{3 x - 1}{x}$

Step 2

Multiply $\frac{4 x + 1}{6 x} \cdot \frac{3 x - 1}{x}$ .

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Step 2.1

Multiply $\frac{4 x + 1}{6 x}$ by $\frac{3 x - 1}{x}$ .

$\frac{(4 x + 1) (3 x - 1)}{6 x \cdot x}$

Step 2.2

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

$\frac{(4 x + 1) (3 x - 1)}{6 (x^{1} x)}$

Step 2.3

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

$\frac{(4 x + 1) (3 x - 1)}{6 (x^{1} x^{1})}$

Step 2.4

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

$\frac{(4 x + 1) (3 x - 1)}{6 x^{1 + 1}}$

Step 2.5

Add $1$ and $1$ .

$\frac{(4 x + 1) (3 x - 1)}{6 x^{2}}$

Step 3

Expand $(4 x + 1) (3 x - 1)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{4 x (3 x - 1) + 1 (3 x - 1)}{6 x^{2}}$

Step 3.2

Apply the distributive property.

$\frac{4 x (3 x) + 4 x \cdot - 1 + 1 (3 x - 1)}{6 x^{2}}$

Step 3.3

Apply the distributive property.

$\frac{4 x (3 x) + 4 x \cdot - 1 + 1 (3 x) + 1 \cdot - 1}{6 x^{2}}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Rewrite using the commutative property of multiplication.

$\frac{4 \cdot 3 x \cdot x + 4 x \cdot - 1 + 1 (3 x) + 1 \cdot - 1}{6 x^{2}}$

Step 4.1.2

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

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Step 4.1.2.1

Move $x$ .

$\frac{4 \cdot 3 (x \cdot x) + 4 x \cdot - 1 + 1 (3 x) + 1 \cdot - 1}{6 x^{2}}$

Step 4.1.2.2

Multiply $x$ by $x$ .

$\frac{4 \cdot 3 x^{2} + 4 x \cdot - 1 + 1 (3 x) + 1 \cdot - 1}{6 x^{2}}$

Step 4.1.3

Multiply $4$ by $3$ .

$\frac{12 x^{2} + 4 x \cdot - 1 + 1 (3 x) + 1 \cdot - 1}{6 x^{2}}$

Step 4.1.4

Multiply $- 1$ by $4$ .

$\frac{12 x^{2} - 4 x + 1 (3 x) + 1 \cdot - 1}{6 x^{2}}$

Step 4.1.5

Multiply $3 x$ by $1$ .

$\frac{12 x^{2} - 4 x + 3 x + 1 \cdot - 1}{6 x^{2}}$

Step 4.1.6

Multiply $- 1$ by $1$ .

$\frac{12 x^{2} - 4 x + 3 x - 1}{6 x^{2}}$

Step 4.2

Add $- 4 x$ and $3 x$ .

$\frac{12 x^{2} - x - 1}{6 x^{2}}$

Step 5

Split the fraction $\frac{12 x^{2} - x - 1}{6 x^{2}}$ into two fractions.

$\frac{12 x^{2} - x}{6 x^{2}} + \frac{- 1}{6 x^{2}}$

Step 6

Split the fraction $\frac{12 x^{2} - x}{6 x^{2}}$ into two fractions.

$\frac{12 x^{2}}{6 x^{2}} + \frac{- x}{6 x^{2}} + \frac{- 1}{6 x^{2}}$

Step 7

Cancel the common factor of $12$ and $6$ .

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Step 7.1

Factor $6$ out of $12 x^{2}$ .

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

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $6$ out of $6 x^{2}$ .

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

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

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

Step 8

Cancel the common factor of $x^{2}$ .

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Step 8.1

Cancel the common factor.

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

Step 8.2

Divide $2$ by $1$ .

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

Step 9

Cancel the common factor of $x$ and $x^{2}$ .

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Step 9.1

Factor $x$ out of $- x$ .

$2 + \frac{x \cdot - 1}{6 x^{2}} + \frac{- 1}{6 x^{2}}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $x$ out of $6 x^{2}$ .

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

Step 9.2.2

Cancel the common factor.

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

Step 9.2.3

Rewrite the expression.

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

Step 10

Move the negative in front of the fraction.

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

Step 11

Move the negative in front of the fraction.

$2 - \frac{1}{6 x} - \frac{1}{6 x^{2}}$