Precalculus Examples

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

Step 1

Multiply both sides by $3$ .

$(\frac{x}{2} + \frac{1}{6}) \cdot 3 = \frac{x}{3} \cdot 3$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $(\frac{x}{2} + \frac{1}{6}) \cdot 3$ .

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Step 2.1.1.1

Simplify terms.

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Step 2.1.1.1.1

Apply the distributive property.

$\frac{x}{2} \cdot 3 + \frac{1}{6} \cdot 3 = \frac{x}{3} \cdot 3$

Step 2.1.1.1.2

Combine $\frac{x}{2}$ and $3$ .

$\frac{x \cdot 3}{2} + \frac{1}{6} \cdot 3 = \frac{x}{3} \cdot 3$

Step 2.1.1.1.3

Cancel the common factor of $3$ .

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Step 2.1.1.1.3.1

Factor $3$ out of $6$ .

$\frac{x \cdot 3}{2} + \frac{1}{3 (2)} \cdot 3 = \frac{x}{3} \cdot 3$

Step 2.1.1.1.3.2

Cancel the common factor.

$\frac{x \cdot 3}{2} + \frac{1}{3 \cdot 2} \cdot 3 = \frac{x}{3} \cdot 3$

Step 2.1.1.1.3.3

Rewrite the expression.

$\frac{x \cdot 3}{2} + \frac{1}{2} = \frac{x}{3} \cdot 3$

Step 2.1.1.2

Move $3$ to the left of $x$ .

$\frac{3 x}{2} + \frac{1}{2} = \frac{x}{3} \cdot 3$

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 x}{2} + \frac{1}{2} = \frac{x}{3} \cdot 3$

Step 2.2.1.2

Rewrite the expression.

$\frac{3 x}{2} + \frac{1}{2} = x$

Step 3

Solve for $x$ .

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Step 3.1

Move all terms containing $x$ to the left side of the equation.

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Step 3.1.1

Subtract $x$ from both sides of the equation.

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.1.4

Combine the numerators over the common denominator.

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

Step 3.1.5

Combine the numerators over the common denominator.

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

Step 3.1.6

Multiply $2$ by $- 1$ .

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

Step 3.1.7

Subtract $2 x$ from $3 x$ .

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

Step 3.2

Set the numerator equal to zero.

$x + 1 = 0$

Step 3.3

Subtract $1$ from both sides of the equation.

$x = - 1$