Solve by Factoring 2/(3x)+1/4=25/(6x)-1/3

$\frac{2}{3 x} + \frac{1}{4} = \frac{25}{6 x} - \frac{1}{3}$

Step 1

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

Subtract $\frac{25}{6 x}$ from both sides of the equation.

$\frac{2}{3 x} + \frac{1}{4} - \frac{25}{6 x} = - \frac{1}{3}$

Add $\frac{1}{3}$ to both sides of the equation.

$\frac{2}{3 x} + \frac{1}{4} - \frac{25}{6 x} + \frac{1}{3} = 0$

Step 2

Simplify $\frac{2}{3 x} + \frac{1}{4} - \frac{25}{6 x} + \frac{1}{3}$ .

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

$\frac{2}{3 x} \cdot \frac{2}{2} - \frac{25}{6 x} + \frac{1}{4} + \frac{1}{3} = 0$

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

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

$\frac{2 \cdot 2}{3 x \cdot 2} - \frac{25}{6 x} + \frac{1}{4} + \frac{1}{3} = 0$

Multiply $2$ by $3$ .

$\frac{2 \cdot 2}{6 x} - \frac{25}{6 x} + \frac{1}{4} + \frac{1}{3} = 0$

Combine the numerators over the common denominator.

$\frac{2 \cdot 2 - 25}{6 x} + \frac{1}{4} + \frac{1}{3} = 0$

Simplify the numerator.

Multiply $2$ by $2$ .

$\frac{4 - 25}{6 x} + \frac{1}{4} + \frac{1}{3} = 0$

Subtract $25$ from $4$ .

$\frac{- 21}{6 x} + \frac{1}{4} + \frac{1}{3} = 0$

Simplify each term.

Cancel the common factor of $- 21$ and $6$ .

Factor $3$ out of $- 21$ .

$\frac{3 (- 7)}{6 x} + \frac{1}{4} + \frac{1}{3} = 0$

Cancel the common factors.

Factor $3$ out of $6 x$ .

$\frac{3 (- 7)}{3 (2 x)} + \frac{1}{4} + \frac{1}{3} = 0$

Cancel the common factor.

$\frac{3 \cdot - 7}{3 (2 x)} + \frac{1}{4} + \frac{1}{3} = 0$

Rewrite the expression.

$\frac{- 7}{2 x} + \frac{1}{4} + \frac{1}{3} = 0$

Move the negative in front of the fraction.

$- \frac{7}{2 x} + \frac{1}{4} + \frac{1}{3} = 0$

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{7}{2 x} + \frac{1}{4} \cdot \frac{3}{3} + \frac{1}{3} = 0$

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{7}{2 x} + \frac{1}{4} \cdot \frac{3}{3} + \frac{1}{3} \cdot \frac{4}{4} = 0$

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{4}$ by $\frac{3}{3}$ .

$- \frac{7}{2 x} + \frac{3}{4 \cdot 3} + \frac{1}{3} \cdot \frac{4}{4} = 0$

Multiply $4$ by $3$ .

$- \frac{7}{2 x} + \frac{3}{12} + \frac{1}{3} \cdot \frac{4}{4} = 0$

Multiply $\frac{1}{3}$ by $\frac{4}{4}$ .

$- \frac{7}{2 x} + \frac{3}{12} + \frac{4}{3 \cdot 4} = 0$

Multiply $3$ by $4$ .

$- \frac{7}{2 x} + \frac{3}{12} + \frac{4}{12} = 0$

Combine the numerators over the common denominator.

$- \frac{7}{2 x} + \frac{3 + 4}{12} = 0$

Add $3$ and $4$ .

$- \frac{7}{2 x} + \frac{7}{12} = 0$

Step 3

To write $- \frac{7}{2 x}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{7}{2 x} \cdot \frac{6}{6} + \frac{7}{12} = 0$

Step 4

To write $\frac{7}{12}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$- \frac{7}{2 x} \cdot \frac{6}{6} + \frac{7}{12} \cdot \frac{x}{x} = 0$

Step 5

Write each expression with a common denominator of $12 x$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{7}{2 x}$ by $\frac{6}{6}$ .

$- \frac{7 \cdot 6}{2 x \cdot 6} + \frac{7}{12} \cdot \frac{x}{x} = 0$

Multiply $6$ by $2$ .

$- \frac{7 \cdot 6}{12 x} + \frac{7}{12} \cdot \frac{x}{x} = 0$

Multiply $\frac{7}{12}$ by $\frac{x}{x}$ .

$- \frac{7 \cdot 6}{12 x} + \frac{7 x}{12 x} = 0$

Step 6

Combine the numerators over the common denominator.

$\frac{- 7 \cdot 6 + 7 x}{12 x} = 0$

Step 7

Simplify the numerator.

Factor $7$ out of $- 7 \cdot 6 + 7 x$ .

Factor $7$ out of $- 7 \cdot 6$ .

$\frac{7 (- 1 \cdot 6) + 7 x}{12 x} = 0$

Factor $7$ out of $7 x$ .

$\frac{7 (- 1 \cdot 6) + 7 (x)}{12 x} = 0$

Factor $7$ out of $7 (- 1 \cdot 6) + 7 (x)$ .

$\frac{7 (- 1 \cdot 6 + x)}{12 x} = 0$

Multiply $- 1$ by $6$ .

$\frac{7 (- 6 + x)}{12 x} = 0$

Step 8

Set the numerator equal to zero.

$7 (- 6 + x) = 0$

Step 9

Solve the equation for $x$ .

Divide each term in $7 (- 6 + x) = 0$ by $7$ and simplify.

Divide each term in $7 (- 6 + x) = 0$ by $7$ .

$\frac{7 (- 6 + x)}{7} = \frac{0}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 (- 6 + x)}{7} = \frac{0}{7}$

Divide $- 6 + x$ by $1$ .

$- 6 + x = \frac{0}{7}$

Simplify the right side.

Divide $0$ by $7$ .

$- 6 + x = 0$

Add $6$ to both sides of the equation.

$x = 6$

Solve by Factoring 2/(3x)+1/4=25/(6x)-1/3

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