Solve by Factoring (4x)/(x-3)+x/2=12/(x-3)

$\frac{4 x}{x - 3} + \frac{x}{2} = \frac{12}{x - 3}$

Step 1

Subtract $\frac{12}{x - 3}$ from both sides of the equation.

$\frac{4 x}{x - 3} + \frac{x}{2} - \frac{12}{x - 3} = 0$

Step 2

Simplify $\frac{4 x}{x - 3} + \frac{x}{2} - \frac{12}{x - 3}$ .

To write $\frac{4 x}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{4 x}{x - 3} \cdot \frac{2}{2} + \frac{x}{2} - \frac{12}{x - 3} = 0$

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

$\frac{4 x}{x - 3} \cdot \frac{2}{2} + \frac{x}{2} \cdot \frac{x - 3}{x - 3} - \frac{12}{x - 3} = 0$

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

Multiply $\frac{4 x}{x - 3}$ by $\frac{2}{2}$ .

$\frac{4 x \cdot 2}{(x - 3) \cdot 2} + \frac{x}{2} \cdot \frac{x - 3}{x - 3} - \frac{12}{x - 3} = 0$

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

$\frac{4 x \cdot 2}{(x - 3) \cdot 2} + \frac{x (x - 3)}{2 (x - 3)} - \frac{12}{x - 3} = 0$

Reorder the factors of $(x - 3) \cdot 2$ .

$\frac{4 x \cdot 2}{2 (x - 3)} + \frac{x (x - 3)}{2 (x - 3)} - \frac{12}{x - 3} = 0$

Combine the numerators over the common denominator.

$\frac{4 x \cdot 2 + x (x - 3)}{2 (x - 3)} - \frac{12}{x - 3} = 0$

Simplify the numerator.

Factor $x$ out of $4 x \cdot 2 + x (x - 3)$ .

Factor $x$ out of $4 x \cdot 2$ .

$\frac{x (4 \cdot 2) + x (x - 3)}{2 (x - 3)} - \frac{12}{x - 3} = 0$

Factor $x$ out of $x (4 \cdot 2) + x (x - 3)$ .

$\frac{x (4 \cdot 2 + x - 3)}{2 (x - 3)} - \frac{12}{x - 3} = 0$

Multiply $4$ by $2$ .

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

Subtract $3$ from $8$ .

$\frac{x (x + 5)}{2 (x - 3)} - \frac{12}{x - 3} = 0$

To write $- \frac{12}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x (x + 5)}{2 (x - 3)} - \frac{12}{x - 3} \cdot \frac{2}{2} = 0$

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

Multiply $\frac{12}{x - 3}$ by $\frac{2}{2}$ .

$\frac{x (x + 5)}{2 (x - 3)} - \frac{12 \cdot 2}{(x - 3) \cdot 2} = 0$

Reorder the factors of $(x - 3) \cdot 2$ .

$\frac{x (x + 5)}{2 (x - 3)} - \frac{12 \cdot 2}{2 (x - 3)} = 0$

Combine the numerators over the common denominator.

$\frac{x (x + 5) - 12 \cdot 2}{2 (x - 3)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{x \cdot x + x \cdot 5 - 12 \cdot 2}{2 (x - 3)} = 0$

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 5 - 12 \cdot 2}{2 (x - 3)} = 0$

Move $5$ to the left of $x$ .

$\frac{x^{2} + 5 \cdot x - 12 \cdot 2}{2 (x - 3)} = 0$

Multiply $- 12$ by $2$ .

$\frac{x^{2} + 5 x - 24}{2 (x - 3)} = 0$

Factor $x^{2} + 5 x - 24$ 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 $- 24$ and whose sum is $5$ .

$- 3, 8$

Write the factored form using these integers.

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

Cancel the common factor of $x - 3$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{x + 8}{2} = 0$

Step 3

Set the numerator equal to zero.

$x + 8 = 0$

Step 4

Subtract $8$ from both sides of the equation.

$x = - 8$

Solve by Factoring (4x)/(x-3)+x/2=12/(x-3)

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