Solve by Factoring 3/(x-1)+8/x=3

$\frac{3}{x - 1} + \frac{8}{x} = 3$

Step 1

Subtract $3$ from both sides of the equation.

$\frac{3}{x - 1} + \frac{8}{x} - 3 = 0$

Step 2

Simplify $\frac{3}{x - 1} + \frac{8}{x} - 3$ .

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

$\frac{3}{x - 1} \cdot \frac{x}{x} + \frac{8}{x} - 3 = 0$

To write $\frac{8}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{3}{x - 1} \cdot \frac{x}{x} + \frac{8}{x} \cdot \frac{x - 1}{x - 1} - 3 = 0$

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

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

$\frac{3 x}{(x - 1) x} + \frac{8}{x} \cdot \frac{x - 1}{x - 1} - 3 = 0$

Multiply $\frac{8}{x}$ by $\frac{x - 1}{x - 1}$ .

$\frac{3 x}{(x - 1) x} + \frac{8 (x - 1)}{x (x - 1)} - 3 = 0$

Reorder the factors of $(x - 1) x$ .

$\frac{3 x}{x (x - 1)} + \frac{8 (x - 1)}{x (x - 1)} - 3 = 0$

Combine the numerators over the common denominator.

$\frac{3 x + 8 (x - 1)}{x (x - 1)} - 3 = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{3 x + 8 x + 8 \cdot - 1}{x (x - 1)} - 3 = 0$

Multiply $8$ by $- 1$ .

$\frac{3 x + 8 x - 8}{x (x - 1)} - 3 = 0$

Add $3 x$ and $8 x$ .

$\frac{11 x - 8}{x (x - 1)} - 3 = 0$

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

$\frac{11 x - 8}{x (x - 1)} - 3 \cdot \frac{x (x - 1)}{x (x - 1)} = 0$

Combine $- 3$ and $\frac{x (x - 1)}{x (x - 1)}$ .

$\frac{11 x - 8}{x (x - 1)} + \frac{- 3 (x (x - 1))}{x (x - 1)} = 0$

Combine the numerators over the common denominator.

$\frac{11 x - 8 - 3 (x (x - 1))}{x (x - 1)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{11 x - 8 - 3 x \cdot x - 3 x \cdot - 1}{x (x - 1)} = 0$

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

Move $x$ .

$\frac{11 x - 8 - 3 (x \cdot x) - 3 x \cdot - 1}{x (x - 1)} = 0$

Multiply $x$ by $x$ .

$\frac{11 x - 8 - 3 x^{2} - 3 x \cdot - 1}{x (x - 1)} = 0$

Multiply $- 1$ by $- 3$ .

$\frac{11 x - 8 - 3 x^{2} + 3 x}{x (x - 1)} = 0$

Add $11 x$ and $3 x$ .

$\frac{14 x - 8 - 3 x^{2}}{x (x - 1)} = 0$

Reorder terms.

$\frac{- 3 x^{2} + 14 x - 8}{x (x - 1)} = 0$

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 $a \cdot c = - 3 \cdot - 8 = 24$ and whose sum is $b = 14$ .

Factor $14$ out of $14 x$ .

$\frac{- 3 x^{2} + 14 (x) - 8}{x (x - 1)} = 0$

Rewrite $14$ as $2$ plus $12$

$\frac{- 3 x^{2} + (2 + 12) x - 8}{x (x - 1)} = 0$

Apply the distributive property.

$\frac{- 3 x^{2} + 2 x + 12 x - 8}{x (x - 1)} = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(- 3 x^{2} + 2 x) + 12 x - 8}{x (x - 1)} = 0$

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

$\frac{x (- 3 x + 2) - 4 (- 3 x + 2)}{x (x - 1)} = 0$

Factor the polynomial by factoring out the greatest common factor, $- 3 x + 2$ .

$\frac{(- 3 x + 2) (x - 4)}{x (x - 1)} = 0$

Factor $- 1$ out of $- 3 x$ .

$\frac{(- (3 x) + 2) (x - 4)}{x (x - 1)} = 0$

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{(- (3 x) - 1 (- 2)) (x - 4)}{x (x - 1)} = 0$

Factor $- 1$ out of $- (3 x) - 1 (- 2)$ .

$\frac{- (3 x - 2) (x - 4)}{x (x - 1)} = 0$

Rewrite $- (3 x - 2)$ as $- 1 (3 x - 2)$ .

$\frac{- 1 (3 x - 2) (x - 4)}{x (x - 1)} = 0$

Move the negative in front of the fraction.

$- \frac{(3 x - 2) (x - 4)}{x (x - 1)} = 0$

Step 3

Set the numerator equal to zero.

$(3 x - 2) (x - 4) = 0$

Step 4

Solve the equation for $x$ .

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$3 x - 2 = 0$

$x - 4 = 0$

Set $3 x - 2$ equal to $0$ and solve for $x$ .

Set $3 x - 2$ equal to $0$ .

$3 x - 2 = 0$

Solve $3 x - 2 = 0$ for $x$ .

Add $2$ to both sides of the equation.

$3 x = 2$

Divide each term in $3 x = 2$ by $3$ and simplify.

Divide each term in $3 x = 2$ by $3$ .

$\frac{3 x}{3} = \frac{2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{2}{3}$

Divide $x$ by $1$ .

$x = \frac{2}{3}$

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

The final solution is all the values that make $(3 x - 2) (x - 4) = 0$ true.

$x = \frac{2}{3}, 4$

Solve by Factoring 3/(x-1)+8/x=3

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