Algebra Examples

$\frac{3}{2 x} - \frac{5}{3 x} = 2$

Step 1

Subtract $2$ from both sides of the equation.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

To write $- \frac{5}{3 x}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Multiply $3$ by $2$ .

$\frac{3 \cdot 3}{6 x} - \frac{5}{3 x} \cdot \frac{2}{2} - 2 = 0$

Step 2.3.3

Multiply $\frac{5}{3 x}$ by $\frac{2}{2}$ .

$\frac{3 \cdot 3}{6 x} - \frac{5 \cdot 2}{3 x \cdot 2} - 2 = 0$

Step 2.3.4

Multiply $2$ by $3$ .

$\frac{3 \cdot 3}{6 x} - \frac{5 \cdot 2}{6 x} - 2 = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{3 \cdot 3 - 5 \cdot 2}{6 x} - 2 = 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $3$ by $3$ .

$\frac{9 - 5 \cdot 2}{6 x} - 2 = 0$

Step 2.5.2

Multiply $- 5$ by $2$ .

$\frac{9 - 10}{6 x} - 2 = 0$

Step 2.5.3

Subtract $10$ from $9$ .

$\frac{- 1}{6 x} - 2 = 0$

Step 2.6

Move the negative in front of the fraction.

$- \frac{1}{6 x} - 2 = 0$

Step 3

Factor $- 1$ out of $- \frac{1}{6 x} - 2$ .

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Step 3.1

Factor $- 1$ out of $- \frac{1}{6 x}$ .

$- (\frac{1}{6 x}) - 2 = 0$

Step 3.2

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

$- (\frac{1}{6 x}) - 1 \cdot 2 = 0$

Step 3.3

Factor $- 1$ out of $- (\frac{1}{6 x}) - 1 (2)$ .

$- (\frac{1}{6 x} + 2) = 0$

Step 4

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

$- (\frac{1}{6 x} + 2 \cdot \frac{6 x}{6 x}) = 0$

Step 5

Combine $2$ and $\frac{6 x}{6 x}$ .

$- (\frac{1}{6 x} + \frac{2 (6 x)}{6 x}) = 0$

Step 6

Combine the numerators over the common denominator.

$- \frac{1 + 2 (6 x)}{6 x} = 0$

Step 7

Remove parentheses.

$- \frac{1 + 2 \cdot (6 x)}{6 x} = 0$

Step 8

Multiply $2$ by $6$ .

$- \frac{1 + 12 x}{6 x} = 0$

Step 9

Set the numerator equal to zero.

$1 + 12 x = 0$

Step 10

Solve the equation for $x$ .

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Step 10.1

Subtract $1$ from both sides of the equation.

$12 x = - 1$

Step 10.2

Divide each term in $12 x = - 1$ by $12$ and simplify.

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Step 10.2.1

Divide each term in $12 x = - 1$ by $12$ .

$\frac{12 x}{12} = \frac{- 1}{12}$

Step 10.2.2

Simplify the left side.

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Step 10.2.2.1

Cancel the common factor of $12$ .

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Step 10.2.2.1.1

Cancel the common factor.

$\frac{12 x}{12} = \frac{- 1}{12}$

Step 10.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{12}$

Step 10.2.3

Simplify the right side.

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Step 10.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{12}$