Solve by Factoring 4/x+9=1/(2x)+16/3

$\frac{4}{x} + 9 = \frac{1}{2 x} + \frac{16}{3}$

Step 1

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

Subtract $\frac{1}{2 x}$ from both sides of the equation.

$\frac{4}{x} + 9 - \frac{1}{2 x} = \frac{16}{3}$

Subtract $\frac{16}{3}$ from both sides of the equation.

$\frac{4}{x} + 9 - \frac{1}{2 x} - \frac{16}{3} = 0$

Step 2

Simplify $\frac{4}{x} + 9 - \frac{1}{2 x} - \frac{16}{3}$ .

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

$\frac{4}{x} \cdot \frac{2}{2} - \frac{1}{2 x} + 9 - \frac{16}{3} = 0$

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

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

$\frac{4 \cdot 2}{x \cdot 2} - \frac{1}{2 x} + 9 - \frac{16}{3} = 0$

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

$\frac{4 \cdot 2}{2 x} - \frac{1}{2 x} + 9 - \frac{16}{3} = 0$

Combine the numerators over the common denominator.

$\frac{4 \cdot 2 - 1}{2 x} + 9 - \frac{16}{3} = 0$

Simplify the numerator.

Multiply $4$ by $2$ .

$\frac{8 - 1}{2 x} + 9 - \frac{16}{3} = 0$

Subtract $1$ from $8$ .

$\frac{7}{2 x} + 9 - \frac{16}{3} = 0$

To write $9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{7}{2 x} + 9 \cdot \frac{3}{3} - \frac{16}{3} = 0$

Combine $9$ and $\frac{3}{3}$ .

$\frac{7}{2 x} + \frac{9 \cdot 3}{3} - \frac{16}{3} = 0$

Combine the numerators over the common denominator.

$\frac{7}{2 x} + \frac{9 \cdot 3 - 16}{3} = 0$

Simplify the numerator.

Multiply $9$ by $3$ .

$\frac{7}{2 x} + \frac{27 - 16}{3} = 0$

Subtract $16$ from $27$ .

$\frac{7}{2 x} + \frac{11}{3} = 0$

Step 3

To write $\frac{7}{2 x}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{7}{2 x} \cdot \frac{3}{3} + \frac{11}{3} = 0$

Step 4

To write $\frac{11}{3}$ as a fraction with a common denominator, multiply by $\frac{2 x}{2 x}$ .

$\frac{7}{2 x} \cdot \frac{3}{3} + \frac{11}{3} \cdot \frac{2 x}{2 x} = 0$

Step 5

Write each expression with a common denominator of $6 x$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{7}{2 x}$ by $\frac{3}{3}$ .

$\frac{7 \cdot 3}{2 x \cdot 3} + \frac{11}{3} \cdot \frac{2 x}{2 x} = 0$

Multiply $3$ by $2$ .

$\frac{7 \cdot 3}{6 x} + \frac{11}{3} \cdot \frac{2 x}{2 x} = 0$

Multiply $\frac{11}{3}$ by $\frac{2 x}{2 x}$ .

$\frac{7 \cdot 3}{6 x} + \frac{11 (2 x)}{3 (2 x)} = 0$

Multiply $2$ by $3$ .

$\frac{7 \cdot 3}{6 x} + \frac{11 (2 x)}{6 x} = 0$

Step 6

Combine the numerators over the common denominator.

$\frac{7 \cdot 3 + 11 (2 x)}{6 x} = 0$

Step 7

Remove parentheses.

$\frac{7 \cdot 3 + 11 \cdot (2 x)}{6 x} = 0$

Step 8

Simplify the numerator.

Multiply $7$ by $3$ .

$\frac{21 + 11 \cdot (2 x)}{6 x} = 0$

Multiply $11$ by $2$ .

$\frac{21 + 22 x}{6 x} = 0$

Step 9

Set the numerator equal to zero.

$21 + 22 x = 0$

Step 10

Solve the equation for $x$ .

Subtract $21$ from both sides of the equation.

$22 x = - 21$

Divide each term in $22 x = - 21$ by $22$ and simplify.

Divide each term in $22 x = - 21$ by $22$ .

$\frac{22 x}{22} = \frac{- 21}{22}$

Simplify the left side.

Cancel the common factor of $22$ .

Cancel the common factor.

$\frac{22 x}{22} = \frac{- 21}{22}$

Divide $x$ by $1$ .

$x = \frac{- 21}{22}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{21}{22}$

Solve by Factoring 4/x+9=1/(2x)+16/3

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