Algebra Examples

$\frac{2 x - 3}{5 x^{2}} \div \frac{x - 5}{3 x - 1}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{2 x - 3}{5 x^{2}} \cdot \frac{3 x - 1}{x - 5}$

Step 2

Multiply $\frac{2 x - 3}{5 x^{2}}$ by $\frac{3 x - 1}{x - 5}$ .

$\frac{(2 x - 3) (3 x - 1)}{5 x^{2} (x - 5)}$

Step 3

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

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Step 3.1

Apply the distributive property.

$\frac{2 x (3 x - 1) - 3 (3 x - 1)}{5 x^{2} (x - 5)}$

Step 3.2

Apply the distributive property.

$\frac{2 x (3 x) + 2 x \cdot - 1 - 3 (3 x - 1)}{5 x^{2} (x - 5)}$

Step 3.3

Apply the distributive property.

$\frac{2 x (3 x) + 2 x \cdot - 1 - 3 (3 x) - 3 \cdot - 1}{5 x^{2} (x - 5)}$

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.

$\frac{2 \cdot 3 x \cdot x + 2 x \cdot - 1 - 3 (3 x) - 3 \cdot - 1}{5 x^{2} (x - 5)}$

Step 4.1.2

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

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Step 4.1.2.1

Move $x$ .

$\frac{2 \cdot 3 (x \cdot x) + 2 x \cdot - 1 - 3 (3 x) - 3 \cdot - 1}{5 x^{2} (x - 5)}$

Step 4.1.2.2

Multiply $x$ by $x$ .

$\frac{2 \cdot 3 x^{2} + 2 x \cdot - 1 - 3 (3 x) - 3 \cdot - 1}{5 x^{2} (x - 5)}$

Step 4.1.3

Multiply $2$ by $3$ .

$\frac{6 x^{2} + 2 x \cdot - 1 - 3 (3 x) - 3 \cdot - 1}{5 x^{2} (x - 5)}$

Step 4.1.4

Multiply $- 1$ by $2$ .

$\frac{6 x^{2} - 2 x - 3 (3 x) - 3 \cdot - 1}{5 x^{2} (x - 5)}$

Step 4.1.5

Multiply $3$ by $- 3$ .

$\frac{6 x^{2} - 2 x - 9 x - 3 \cdot - 1}{5 x^{2} (x - 5)}$

Step 4.1.6

Multiply $- 3$ by $- 1$ .

$\frac{6 x^{2} - 2 x - 9 x + 3}{5 x^{2} (x - 5)}$

Step 4.2

Subtract $9 x$ from $- 2 x$ .

$\frac{6 x^{2} - 11 x + 3}{5 x^{2} (x - 5)}$

Step 5

Split the fraction $\frac{6 x^{2} - 11 x + 3}{5 x^{2} (x - 5)}$ into two fractions.

$\frac{6 x^{2} - 11 x}{5 x^{2} (x - 5)} + \frac{3}{5 x^{2} (x - 5)}$

Step 6

Split the fraction $\frac{6 x^{2} - 11 x}{5 x^{2} (x - 5)}$ into two fractions.

$\frac{6 x^{2}}{5 x^{2} (x - 5)} + \frac{- 11 x}{5 x^{2} (x - 5)} + \frac{3}{5 x^{2} (x - 5)}$

Step 7

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

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Step 7.1

Cancel the common factor.

$\frac{6 x^{2}}{5 x^{2} (x - 5)} + \frac{- 11 x}{5 x^{2} (x - 5)} + \frac{3}{5 x^{2} (x - 5)}$

Step 7.2

Rewrite the expression.

$\frac{6}{5 (x - 5)} + \frac{- 11 x}{5 x^{2} (x - 5)} + \frac{3}{5 x^{2} (x - 5)}$

Step 8

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

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Step 8.1

Factor $x$ out of $- 11 x$ .

$\frac{6}{5 (x - 5)} + \frac{x \cdot - 11}{5 x^{2} (x - 5)} + \frac{3}{5 x^{2} (x - 5)}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $x$ out of $5 x^{2} (x - 5)$ .

$\frac{6}{5 (x - 5)} + \frac{x \cdot - 11}{x (5 x (x - 5))} + \frac{3}{5 x^{2} (x - 5)}$

Step 8.2.2

Cancel the common factor.

$\frac{6}{5 (x - 5)} + \frac{x \cdot - 11}{x (5 x (x - 5))} + \frac{3}{5 x^{2} (x - 5)}$

Step 8.2.3

Rewrite the expression.

$\frac{6}{5 (x - 5)} + \frac{- 11}{5 x (x - 5)} + \frac{3}{5 x^{2} (x - 5)}$

Step 9

Move the negative in front of the fraction.

$\frac{6}{5 (x - 5)} - \frac{11}{5 x (x - 5)} + \frac{3}{5 x^{2} (x - 5)}$