Algebra Examples

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

Step 1

Simplify $\frac{x}{2} + \frac{x}{3}$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

Multiply $2$ by $3$ .

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

Step 1.3.3

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

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

Step 1.3.4

Multiply $3$ by $2$ .

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 1.5

Simplify the numerator.

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Step 1.5.1

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

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

Step 1.5.2

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

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

Step 1.5.3

Add $3 x$ and $2 x$ .

$\frac{5 x}{6} = x - 2$

Step 2

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

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Step 2.1

Subtract $x$ from both sides of the equation.

$\frac{5 x}{6} - x = - 2$

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Factor $x$ out of $5 x - x \cdot 6$ .

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Step 2.5.1.1

Factor $x$ out of $5 x$ .

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

Step 2.5.1.2

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

$\frac{x \cdot 5 + x (- 1 \cdot 6)}{6} = - 2$

Step 2.5.1.3

Factor $x$ out of $x \cdot 5 + x (- 1 \cdot 6)$ .

$\frac{x (5 - 1 \cdot 6)}{6} = - 2$

Step 2.5.2

Multiply $- 1$ by $6$ .

$\frac{x (5 - 6)}{6} = - 2$

Step 2.5.3

Subtract $6$ from $5$ .

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

Step 2.6

Move $- 1$ to the left of $x$ .

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

Step 2.7

Move the negative in front of the fraction.

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

Step 3

Multiply both sides of the equation by $- 6$ .

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

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $- 6 (- \frac{x}{6})$ .

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Step 4.1.1.1

Cancel the common factor of $6$ .

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Step 4.1.1.1.1

Move the leading negative in $- \frac{x}{6}$ into the numerator.

$- 6 \frac{- x}{6} = - 6 \cdot - 2$

Step 4.1.1.1.2

Factor $6$ out of $- 6$ .

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

Step 4.1.1.1.3

Cancel the common factor.

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

Step 4.1.1.1.4

Rewrite the expression.

$- - x = - 6 \cdot - 2$

Step 4.1.1.2

Multiply.

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Step 4.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 x = - 6 \cdot - 2$

Step 4.1.1.2.2

Multiply $x$ by $1$ .

$x = - 6 \cdot - 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $- 6$ by $- 2$ .

$x = 12$