Algebra Examples

$\frac{x}{5 x - 6} - \frac{x + 3}{6 x}$

Step 1

To write $\frac{x}{5 x - 6}$ as a fraction with a common denominator, multiply by $\frac{6 x}{6 x}$ .

$\frac{x}{5 x - 6} \cdot \frac{6 x}{6 x} - \frac{x + 3}{6 x}$

Step 2

To write $- \frac{x + 3}{6 x}$ as a fraction with a common denominator, multiply by $\frac{5 x - 6}{5 x - 6}$ .

$\frac{x}{5 x - 6} \cdot \frac{6 x}{6 x} - \frac{x + 3}{6 x} \cdot \frac{5 x - 6}{5 x - 6}$

Step 3

Write each expression with a common denominator of $(5 x - 6) \cdot 6 x$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 3.1

Multiply $\frac{x}{5 x - 6}$ by $\frac{6 x}{6 x}$ .

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

Step 3.2

Multiply $\frac{x + 3}{6 x}$ by $\frac{5 x - 6}{5 x - 6}$ .

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

Step 3.3

Reorder the factors of $(5 x - 6) (6 x)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Tap for more steps...

Step 5.1

Rewrite using the commutative property of multiplication.

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

Move $x$ .

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

Step 5.2.2

Multiply $x$ by $x$ .

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

Step 5.3

Apply the distributive property.

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

Step 5.4

Multiply $- 1$ by $3$ .

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

Step 5.5

Expand $(- x - 3) (5 x - 6)$ using the FOIL Method.

Tap for more steps...

Step 5.5.1

Apply the distributive property.

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

Step 5.5.2

Apply the distributive property.

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

Step 5.5.3

Apply the distributive property.

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

Step 5.6

Simplify and combine like terms.

Tap for more steps...

Step 5.6.1

Simplify each term.

Tap for more steps...

Step 5.6.1.1

Rewrite using the commutative property of multiplication.

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

Step 5.6.1.2

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

Tap for more steps...

Step 5.6.1.2.1

Move $x$ .

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

Step 5.6.1.2.2

Multiply $x$ by $x$ .

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

Step 5.6.1.3

Multiply $- 1$ by $5$ .

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

Step 5.6.1.4

Multiply $- 6$ by $- 1$ .

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

Step 5.6.1.5

Multiply $5$ by $- 3$ .

$\frac{6 x^{2} - 5 x^{2} + 6 x - 15 x - 3 \cdot - 6}{6 x (5 x - 6)}$

Step 5.6.1.6

Multiply $- 3$ by $- 6$ .

$\frac{6 x^{2} - 5 x^{2} + 6 x - 15 x + 18}{6 x (5 x - 6)}$

Step 5.6.2

Subtract $15 x$ from $6 x$ .

$\frac{6 x^{2} - 5 x^{2} - 9 x + 18}{6 x (5 x - 6)}$

Step 5.7

Subtract $5 x^{2}$ from $6 x^{2}$ .

$\frac{x^{2} - 9 x + 18}{6 x (5 x - 6)}$

Step 5.8

Factor $x^{2} - 9 x + 18$ using the AC method.

Tap for more steps...

Step 5.8.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $18$ and whose sum is $- 9$ .

$- 6, - 3$

Step 5.8.2

Write the factored form using these integers.

$\frac{(x - 6) (x - 3)}{6 x (5 x - 6)}$