Precalculus Examples

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

Step 1

Factor each term.

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Step 1.1

Factor $2$ out of $2 x + 4$ .

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Step 1.1.1

Factor $2$ out of $2 x$ .

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

Step 1.1.2

Factor $2$ out of $4$ .

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

Step 1.1.3

Factor $2$ out of $2 x + 2 \cdot 2$ .

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

Step 1.2

Reduce the expression by cancelling the common factors.

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Step 1.2.1

Reduce the expression $\frac{2 (x + 2)}{2}$ by cancelling the common factors.

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Step 1.2.1.1

Cancel the common factor.

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

Step 1.2.1.2

Rewrite the expression.

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

Step 1.2.2

Divide $x + 2$ by $1$ .

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

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, 1, x - 2, 1$

Step 2.2

Remove parentheses.

$1, 1, x - 2, 1$

Step 2.3

The LCM of one and any expression is the expression.

$x - 2$

Step 3

Multiply each term in $x + 2 - \frac{3 x}{x - 2} = x$ by $x - 2$ to eliminate the fractions.

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Step 3.1

Multiply each term in $x + 2 - \frac{3 x}{x - 2} = x$ by $x - 2$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Apply the distributive property.

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

Step 3.2.1.2

Multiply $x$ by $x$ .

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

Step 3.2.1.3

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

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

Step 3.2.1.4

Apply the distributive property.

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

Step 3.2.1.5

Multiply $2$ by $- 2$ .

$x^{2} - 2 x + 2 x - 4 - \frac{3 x}{x - 2} (x - 2) = x (x - 2)$

Step 3.2.1.6

Cancel the common factor of $x - 2$ .

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Step 3.2.1.6.1

Move the leading negative in $- \frac{3 x}{x - 2}$ into the numerator.

$x^{2} - 2 x + 2 x - 4 + \frac{- 3 x}{x - 2} (x - 2) = x (x - 2)$

Step 3.2.1.6.2

Cancel the common factor.

$x^{2} - 2 x + 2 x - 4 + \frac{- 3 x}{x - 2} (x - 2) = x (x - 2)$

Step 3.2.1.6.3

Rewrite the expression.

$x^{2} - 2 x + 2 x - 4 - 3 x = x (x - 2)$

Step 3.2.2

Combine the opposite terms in $x^{2} - 2 x + 2 x - 4 - 3 x$ .

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Step 3.2.2.1

Add $- 2 x$ and $2 x$ .

$x^{2} + 0 - 4 - 3 x = x (x - 2)$

Step 3.2.2.2

Add $x^{2}$ and $0$ .

$x^{2} - 4 - 3 x = x (x - 2)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Apply the distributive property.

$x^{2} - 4 - 3 x = x \cdot x + x \cdot - 2$

Step 3.3.2

Simplify the expression.

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Step 3.3.2.1

Multiply $x$ by $x$ .

$x^{2} - 4 - 3 x = x^{2} + x \cdot - 2$

Step 3.3.2.2

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

$x^{2} - 4 - 3 x = x^{2} - 2 x$

Step 4

Solve the equation.

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Step 4.1

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

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Step 4.1.1

Subtract $x^{2}$ from both sides of the equation.

$x^{2} - 4 - 3 x - x^{2} = - 2 x$

Step 4.1.2

Add $2 x$ to both sides of the equation.

$x^{2} - 4 - 3 x - x^{2} + 2 x = 0$

Step 4.1.3

Combine the opposite terms in $x^{2} - 4 - 3 x - x^{2} + 2 x$ .

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Step 4.1.3.1

Subtract $x^{2}$ from $x^{2}$ .

$- 4 - 3 x + 0 + 2 x = 0$

Step 4.1.3.2

Add $- 4 - 3 x$ and $0$ .

$- 4 - 3 x + 2 x = 0$

Step 4.1.4

Add $- 3 x$ and $2 x$ .

$- 4 - x = 0$

Step 4.2

Add $4$ to both sides of the equation.

$- x = 4$

Step 4.3

Divide each term in $- x = 4$ by $- 1$ and simplify.

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Step 4.3.1

Divide each term in $- x = 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{4}{- 1}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{4}{- 1}$

Step 4.3.2.2

Divide $x$ by $1$ .

$x = \frac{4}{- 1}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide $4$ by $- 1$ .

$x = - 4$