Simplify 5/(6x-18)-(x-1)/(4x^2-14x+6)

$\frac{5}{6 x - 18} - \frac{x - 1}{4 x^{2} - 14 x + 6}$

Step 1

Simplify each term.

Factor $6$ out of $6 x - 18$ .

Factor $6$ out of $6 x$ .

$\frac{5}{6 (x) - 18} - \frac{x - 1}{4 x^{2} - 14 x + 6}$

Factor $6$ out of $- 18$ .

$\frac{5}{6 x + 6 \cdot - 3} - \frac{x - 1}{4 x^{2} - 14 x + 6}$

Factor $6$ out of $6 x + 6 \cdot - 3$ .

$\frac{5}{6 (x - 3)} - \frac{x - 1}{4 x^{2} - 14 x + 6}$

Simplify the denominator.

Factor $2$ out of $4 x^{2} - 14 x + 6$ .

Factor $2$ out of $4 x^{2}$ .

$\frac{5}{6 (x - 3)} - \frac{x - 1}{2 (2 x^{2}) - 14 x + 6}$

Factor $2$ out of $- 14 x$ .

$\frac{5}{6 (x - 3)} - \frac{x - 1}{2 (2 x^{2}) + 2 (- 7 x) + 6}$

Factor $2$ out of $6$ .

$\frac{5}{6 (x - 3)} - \frac{x - 1}{2 (2 x^{2}) + 2 (- 7 x) + 2 (3)}$

Factor $2$ out of $2 (2 x^{2}) + 2 (- 7 x)$ .

$\frac{5}{6 (x - 3)} - \frac{x - 1}{2 (2 x^{2} - 7 x) + 2 (3)}$

Factor $2$ out of $2 (2 x^{2} - 7 x) + 2 (3)$ .

$\frac{5}{6 (x - 3)} - \frac{x - 1}{2 (2 x^{2} - 7 x + 3)}$

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 $a \cdot c = 2 \cdot 3 = 6$ and whose sum is $b = - 7$ .

Factor $- 7$ out of $- 7 x$ .

$\frac{5}{6 (x - 3)} - \frac{x - 1}{2 (2 x^{2} - 7 (x) + 3)}$

Rewrite $- 7$ as $- 1$ plus $- 6$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

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

Step 2

To write $\frac{5}{6 (x - 3)}$ as a fraction with a common denominator, multiply by $\frac{2 x - 1}{2 x - 1}$ .

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

Step 3

To write $- \frac{x - 1}{2 (2 x - 1) (x - 3)}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 4

Write each expression with a common denominator of $6 (x - 3) (2 x - 1)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5}{6 (x - 3)}$ by $\frac{2 x - 1}{2 x - 1}$ .

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

Multiply $\frac{x - 1}{2 (2 x - 1) (x - 3)}$ by $\frac{3}{3}$ .

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

Multiply $3$ by $2$ .

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

Reorder the factors of $6 (x - 3) (2 x - 1)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Apply the distributive property.

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

Multiply $2$ by $5$ .

$\frac{10 x + 5 \cdot - 1 - (x - 1) \cdot 3}{6 (2 x - 1) (x - 3)}$

Multiply $5$ by $- 1$ .

$\frac{10 x - 5 - (x - 1) \cdot 3}{6 (2 x - 1) (x - 3)}$

Apply the distributive property.

$\frac{10 x - 5 + (- x - - 1) \cdot 3}{6 (2 x - 1) (x - 3)}$

Multiply $- 1$ by $- 1$ .

$\frac{10 x - 5 + (- x + 1) \cdot 3}{6 (2 x - 1) (x - 3)}$

Apply the distributive property.

$\frac{10 x - 5 - x \cdot 3 + 1 \cdot 3}{6 (2 x - 1) (x - 3)}$

Multiply $3$ by $- 1$ .

$\frac{10 x - 5 - 3 x + 1 \cdot 3}{6 (2 x - 1) (x - 3)}$

Multiply $3$ by $1$ .

$\frac{10 x - 5 - 3 x + 3}{6 (2 x - 1) (x - 3)}$

Subtract $3 x$ from $10 x$ .

$\frac{7 x - 5 + 3}{6 (2 x - 1) (x - 3)}$

Add $- 5$ and $3$ .

$\frac{7 x - 2}{6 (2 x - 1) (x - 3)}$

Simplify 5/(6x-18)-(x-1)/(4x^2-14x+6)

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