Simplify x/(5x-6)-(x-1)/(7x)

$\frac{x}{5 x - 6} - \frac{x - 1}{7 x}$

Step 1

To write $\frac{x}{5 x - 6}$ as a fraction with a common denominator, multiply by $\frac{7 x}{7 x}$ .

$\frac{x}{5 x - 6} \cdot \frac{7 x}{7 x} - \frac{x - 1}{7 x}$

Step 2

To write $- \frac{x - 1}{7 x}$ as a fraction with a common denominator, multiply by $\frac{5 x - 6}{5 x - 6}$ .

$\frac{x}{5 x - 6} \cdot \frac{7 x}{7 x} - \frac{x - 1}{7 x} \cdot \frac{5 x - 6}{5 x - 6}$

Step 3

Write each expression with a common denominator of $(5 x - 6) \cdot 7 x$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x}{5 x - 6}$ by $\frac{7 x}{7 x}$ .

$\frac{x (7 x)}{(5 x - 6) (7 x)} - \frac{x - 1}{7 x} \cdot \frac{5 x - 6}{5 x - 6}$

Multiply $\frac{x - 1}{7 x}$ by $\frac{5 x - 6}{5 x - 6}$ .

$\frac{x (7 x)}{(5 x - 6) (7 x)} - \frac{(x - 1) (5 x - 6)}{7 x (5 x - 6)}$

Reorder the factors of $(5 x - 6) (7 x)$ .

$\frac{x (7 x)}{7 x (5 x - 6)} - \frac{(x - 1) (5 x - 6)}{7 x (5 x - 6)}$

Step 4

Combine the numerators over the common denominator.

$\frac{x (7 x) - (x - 1) (5 x - 6)}{7 x (5 x - 6)}$

Step 5

Simplify the numerator.

Rewrite using the commutative property of multiplication.

$\frac{7 x \cdot x - (x - 1) (5 x - 6)}{7 x (5 x - 6)}$

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

Move $x$ .

$\frac{7 (x \cdot x) - (x - 1) (5 x - 6)}{7 x (5 x - 6)}$

Multiply $x$ by $x$ .

$\frac{7 x^{2} - (x - 1) (5 x - 6)}{7 x (5 x - 6)}$

Apply the distributive property.

$\frac{7 x^{2} + (- x - - 1) (5 x - 6)}{7 x (5 x - 6)}$

Multiply $- 1$ by $- 1$ .

$\frac{7 x^{2} + (- x + 1) (5 x - 6)}{7 x (5 x - 6)}$

Expand $(- x + 1) (5 x - 6)$ using the FOIL Method.

Apply the distributive property.

$\frac{7 x^{2} - x (5 x - 6) + 1 (5 x - 6)}{7 x (5 x - 6)}$

Apply the distributive property.

$\frac{7 x^{2} - x (5 x) - x \cdot - 6 + 1 (5 x - 6)}{7 x (5 x - 6)}$

Apply the distributive property.

$\frac{7 x^{2} - x (5 x) - x \cdot - 6 + 1 (5 x) + 1 \cdot - 6}{7 x (5 x - 6)}$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$\frac{7 x^{2} - 1 \cdot 5 x \cdot x - x \cdot - 6 + 1 (5 x) + 1 \cdot - 6}{7 x (5 x - 6)}$

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

Move $x$ .

$\frac{7 x^{2} - 1 \cdot 5 (x \cdot x) - x \cdot - 6 + 1 (5 x) + 1 \cdot - 6}{7 x (5 x - 6)}$

Multiply $x$ by $x$ .

$\frac{7 x^{2} - 1 \cdot 5 x^{2} - x \cdot - 6 + 1 (5 x) + 1 \cdot - 6}{7 x (5 x - 6)}$

Multiply $- 1$ by $5$ .

$\frac{7 x^{2} - 5 x^{2} - x \cdot - 6 + 1 (5 x) + 1 \cdot - 6}{7 x (5 x - 6)}$

Multiply $- 6$ by $- 1$ .

$\frac{7 x^{2} - 5 x^{2} + 6 x + 1 (5 x) + 1 \cdot - 6}{7 x (5 x - 6)}$

Multiply $5 x$ by $1$ .

$\frac{7 x^{2} - 5 x^{2} + 6 x + 5 x + 1 \cdot - 6}{7 x (5 x - 6)}$

Multiply $- 6$ by $1$ .

$\frac{7 x^{2} - 5 x^{2} + 6 x + 5 x - 6}{7 x (5 x - 6)}$

Add $6 x$ and $5 x$ .

$\frac{7 x^{2} - 5 x^{2} + 11 x - 6}{7 x (5 x - 6)}$

Subtract $5 x^{2}$ from $7 x^{2}$ .

$\frac{2 x^{2} + 11 x - 6}{7 x (5 x - 6)}$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 6 = - 12$ and whose sum is $b = 11$ .

Factor $11$ out of $11 x$ .

$\frac{2 x^{2} + 11 (x) - 6}{7 x (5 x - 6)}$

Rewrite $11$ as $- 1$ plus $12$

$\frac{2 x^{2} + (- 1 + 12) x - 6}{7 x (5 x - 6)}$

Apply the distributive property.

$\frac{2 x^{2} - 1 x + 12 x - 6}{7 x (5 x - 6)}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(2 x^{2} - 1 x) + 12 x - 6}{7 x (5 x - 6)}$

Factor out the greatest common factor (GCF) from each group.

$\frac{x (2 x - 1) + 6 (2 x - 1)}{7 x (5 x - 6)}$

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$\frac{(2 x - 1) (x + 6)}{7 x (5 x - 6)}$

Simplify x/(5x-6)-(x-1)/(7x)

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