Algebra Examples

$\frac{p}{3 p - 1} - \frac{p}{3 p + 1}$

Step 1

To write $\frac{p}{3 p - 1}$ as a fraction with a common denominator, multiply by $\frac{3 p + 1}{3 p + 1}$ .

$\frac{p}{3 p - 1} \cdot \frac{3 p + 1}{3 p + 1} - \frac{p}{3 p + 1}$

Step 2

To write $- \frac{p}{3 p + 1}$ as a fraction with a common denominator, multiply by $\frac{3 p - 1}{3 p - 1}$ .

$\frac{p}{3 p - 1} \cdot \frac{3 p + 1}{3 p + 1} - \frac{p}{3 p + 1} \cdot \frac{3 p - 1}{3 p - 1}$

Step 3

Write each expression with a common denominator of $(3 p - 1) (3 p + 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{p}{3 p - 1}$ by $\frac{3 p + 1}{3 p + 1}$ .

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

Step 3.2

Multiply $\frac{p}{3 p + 1}$ by $\frac{3 p - 1}{3 p - 1}$ .

$\frac{p (3 p + 1)}{(3 p - 1) (3 p + 1)} - \frac{p (3 p - 1)}{(3 p + 1) (3 p - 1)}$

Step 3.3

Reorder the factors of $(3 p - 1) (3 p + 1)$ .

$\frac{p (3 p + 1)}{(3 p + 1) (3 p - 1)} - \frac{p (3 p - 1)}{(3 p + 1) (3 p - 1)}$

Step 4

Combine the numerators over the common denominator.

$\frac{p (3 p + 1) - p (3 p - 1)}{(3 p + 1) (3 p - 1)}$

Step 5

Simplify the numerator.

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Step 5.1

Factor $p$ out of $p (3 p + 1) - p (3 p - 1)$ .

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Step 5.1.1

Factor $p$ out of $- p (3 p - 1)$ .

$\frac{p (3 p + 1) + p (- (3 p - 1))}{(3 p + 1) (3 p - 1)}$

Step 5.1.2

Factor $p$ out of $p (3 p + 1) + p (- (3 p - 1))$ .

$\frac{p (3 p + 1 - (3 p - 1))}{(3 p + 1) (3 p - 1)}$

Step 5.2

Apply the distributive property.

$\frac{p (3 p + 1 - (3 p) - - 1)}{(3 p + 1) (3 p - 1)}$

Step 5.3

Multiply $3$ by $- 1$ .

$\frac{p (3 p + 1 - 3 p - - 1)}{(3 p + 1) (3 p - 1)}$

Step 5.4

Multiply $- 1$ by $- 1$ .

$\frac{p (3 p + 1 - 3 p + 1)}{(3 p + 1) (3 p - 1)}$

Step 5.5

Subtract $3 p$ from $3 p$ .

$\frac{p (0 + 1 + 1)}{(3 p + 1) (3 p - 1)}$

Step 5.6

Add $0$ and $1$ .

$\frac{p (1 + 1)}{(3 p + 1) (3 p - 1)}$

Step 5.7

Add $1$ and $1$ .

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

Step 6

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

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