Solve by Factoring 2/x+3=5/(2x)+13/4

$\frac{2}{x} + 3 = \frac{5}{2 x} + \frac{13}{4}$

Step 1

Move all the expressions to the left side of the equation.

Subtract $\frac{5}{2 x}$ from both sides of the equation.

$\frac{2}{x} + 3 - \frac{5}{2 x} = \frac{13}{4}$

Subtract $\frac{13}{4}$ from both sides of the equation.

$\frac{2}{x} + 3 - \frac{5}{2 x} - \frac{13}{4} = 0$

Step 2

Simplify $\frac{2}{x} + 3 - \frac{5}{2 x} - \frac{13}{4}$ .

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

$\frac{2}{x} \cdot \frac{2}{2} - \frac{5}{2 x} + 3 - \frac{13}{4} = 0$

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

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

$\frac{2 \cdot 2}{x \cdot 2} - \frac{5}{2 x} + 3 - \frac{13}{4} = 0$

Move $2$ to the left of $x$ .

$\frac{2 \cdot 2}{2 x} - \frac{5}{2 x} + 3 - \frac{13}{4} = 0$

Combine the numerators over the common denominator.

$\frac{2 \cdot 2 - 5}{2 x} + 3 - \frac{13}{4} = 0$

Simplify the numerator.

Multiply $2$ by $2$ .

$\frac{4 - 5}{2 x} + 3 - \frac{13}{4} = 0$

Subtract $5$ from $4$ .

$\frac{- 1}{2 x} + 3 - \frac{13}{4} = 0$

Move the negative in front of the fraction.

$- \frac{1}{2 x} + 3 - \frac{13}{4} = 0$

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

$- \frac{1}{2 x} + 3 \cdot \frac{4}{4} - \frac{13}{4} = 0$

Combine $3$ and $\frac{4}{4}$ .

$- \frac{1}{2 x} + \frac{3 \cdot 4}{4} - \frac{13}{4} = 0$

Combine the numerators over the common denominator.

$- \frac{1}{2 x} + \frac{3 \cdot 4 - 13}{4} = 0$

Simplify the numerator.

Multiply $3$ by $4$ .

$- \frac{1}{2 x} + \frac{12 - 13}{4} = 0$

Subtract $13$ from $12$ .

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

Move the negative in front of the fraction.

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

Step 3

Factor $- 1$ out of $- \frac{1}{2 x} - \frac{1}{4}$ .

Factor $- 1$ out of $- \frac{1}{2 x}$ .

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

Factor $- 1$ out of $- \frac{1}{4}$ .

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

Factor $- 1$ out of $- (\frac{1}{2 x}) - (\frac{1}{4})$ .

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

Step 4

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

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

Step 5

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 6

Write each expression with a common denominator of $4 x$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{2 x}$ by $\frac{2}{2}$ .

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

Multiply $2$ by $2$ .

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

Multiply $\frac{1}{4}$ by $\frac{x}{x}$ .

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Set the numerator equal to zero.

$2 + x = 0$

Step 9

Subtract $2$ from both sides of the equation.

$x = - 2$

Solve by Factoring 2/x+3=5/(2x)+13/4

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