Simplify x/(8x-7)-(x+2)/(9x)

$\frac{x}{8 x - 7} - \frac{x + 2}{9 x}$

Step 1

To write $\frac{x}{8 x - 7}$ as a fraction with a common denominator, multiply by $\frac{9 x}{9 x}$ .

$\frac{x}{8 x - 7} \cdot \frac{9 x}{9 x} - \frac{x + 2}{9 x}$

Step 2

To write $- \frac{x + 2}{9 x}$ as a fraction with a common denominator, multiply by $\frac{8 x - 7}{8 x - 7}$ .

$\frac{x}{8 x - 7} \cdot \frac{9 x}{9 x} - \frac{x + 2}{9 x} \cdot \frac{8 x - 7}{8 x - 7}$

Step 3

Write each expression with a common denominator of $(8 x - 7) \cdot 9 x$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x}{8 x - 7}$ by $\frac{9 x}{9 x}$ .

$\frac{x (9 x)}{(8 x - 7) (9 x)} - \frac{x + 2}{9 x} \cdot \frac{8 x - 7}{8 x - 7}$

Multiply $\frac{x + 2}{9 x}$ by $\frac{8 x - 7}{8 x - 7}$ .

$\frac{x (9 x)}{(8 x - 7) (9 x)} - \frac{(x + 2) (8 x - 7)}{9 x (8 x - 7)}$

Reorder the factors of $(8 x - 7) (9 x)$ .

$\frac{x (9 x)}{9 x (8 x - 7)} - \frac{(x + 2) (8 x - 7)}{9 x (8 x - 7)}$

Step 4

Combine the numerators over the common denominator.

$\frac{x (9 x) - (x + 2) (8 x - 7)}{9 x (8 x - 7)}$

Step 5

Simplify the numerator.

Rewrite using the commutative property of multiplication.

$\frac{9 x \cdot x - (x + 2) (8 x - 7)}{9 x (8 x - 7)}$

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

Move $x$ .

$\frac{9 (x \cdot x) - (x + 2) (8 x - 7)}{9 x (8 x - 7)}$

Multiply $x$ by $x$ .

$\frac{9 x^{2} - (x + 2) (8 x - 7)}{9 x (8 x - 7)}$

Apply the distributive property.

$\frac{9 x^{2} + (- x - 1 \cdot 2) (8 x - 7)}{9 x (8 x - 7)}$

Multiply $- 1$ by $2$ .

$\frac{9 x^{2} + (- x - 2) (8 x - 7)}{9 x (8 x - 7)}$

Expand $(- x - 2) (8 x - 7)$ using the FOIL Method.

Apply the distributive property.

$\frac{9 x^{2} - x (8 x - 7) - 2 (8 x - 7)}{9 x (8 x - 7)}$

Apply the distributive property.

$\frac{9 x^{2} - x (8 x) - x \cdot - 7 - 2 (8 x - 7)}{9 x (8 x - 7)}$

Apply the distributive property.

$\frac{9 x^{2} - x (8 x) - x \cdot - 7 - 2 (8 x) - 2 \cdot - 7}{9 x (8 x - 7)}$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$\frac{9 x^{2} - 1 \cdot 8 x \cdot x - x \cdot - 7 - 2 (8 x) - 2 \cdot - 7}{9 x (8 x - 7)}$

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

Move $x$ .

$\frac{9 x^{2} - 1 \cdot 8 (x \cdot x) - x \cdot - 7 - 2 (8 x) - 2 \cdot - 7}{9 x (8 x - 7)}$

Multiply $x$ by $x$ .

$\frac{9 x^{2} - 1 \cdot 8 x^{2} - x \cdot - 7 - 2 (8 x) - 2 \cdot - 7}{9 x (8 x - 7)}$

Multiply $- 1$ by $8$ .

$\frac{9 x^{2} - 8 x^{2} - x \cdot - 7 - 2 (8 x) - 2 \cdot - 7}{9 x (8 x - 7)}$

Multiply $- 7$ by $- 1$ .

$\frac{9 x^{2} - 8 x^{2} + 7 x - 2 (8 x) - 2 \cdot - 7}{9 x (8 x - 7)}$

Multiply $8$ by $- 2$ .

$\frac{9 x^{2} - 8 x^{2} + 7 x - 16 x - 2 \cdot - 7}{9 x (8 x - 7)}$

Multiply $- 2$ by $- 7$ .

$\frac{9 x^{2} - 8 x^{2} + 7 x - 16 x + 14}{9 x (8 x - 7)}$

Subtract $16 x$ from $7 x$ .

$\frac{9 x^{2} - 8 x^{2} - 9 x + 14}{9 x (8 x - 7)}$

Subtract $8 x^{2}$ from $9 x^{2}$ .

$\frac{x^{2} - 9 x + 14}{9 x (8 x - 7)}$

Factor $x^{2} - 9 x + 14$ using the AC method.

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 $14$ and whose sum is $- 9$ .

$- 7, - 2$

Write the factored form using these integers.

$\frac{(x - 7) (x - 2)}{9 x (8 x - 7)}$

Simplify x/(8x-7)-(x+2)/(9x)

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