Algebra Examples

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

Step 1

Subtract $\frac{x - 2}{2}$ from both sides of the equation.

$\frac{2 x - 4}{x} - \frac{x - 2}{2} = 0$

Step 2

Simplify $\frac{2 x - 4}{x} - \frac{x - 2}{2}$ .

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Step 2.1

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

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Step 2.1.1

Factor $2$ out of $2 x$ .

$\frac{2 (x) - 4}{x} - \frac{x - 2}{2} = 0$

Step 2.1.2

Factor $2$ out of $- 4$ .

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

Step 2.1.3

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Write each expression with a common denominator of $x \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{2 (x - 2)}{x}$ by $\frac{2}{2}$ .

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

Step 2.4.2

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

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

Step 2.4.3

Reorder the factors of $x \cdot 2$ .

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Factor $x - 2$ out of $2 (x - 2) \cdot 2 - (x - 2) x$ .

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Step 2.6.1.1

Factor $x - 2$ out of $2 (x - 2) \cdot 2$ .

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

Step 2.6.1.2

Factor $x - 2$ out of $- (x - 2) x$ .

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

Step 2.6.1.3

Factor $x - 2$ out of $(x - 2) (2 \cdot 2) + (x - 2) (- 1 x)$ .

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

Step 2.6.2

Multiply $2$ by $2$ .

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

Step 2.6.3

Rewrite $- 1 x$ as $- x$ .

$\frac{(x - 2) (4 - x)}{2 x} = 0$

Step 3

Set the numerator equal to zero.

$(x - 2) (4 - x) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 2 = 0$

$4 - x = 0$

Step 4.2

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 4.2.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 4.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 4.3

Set $4 - x$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $4 - x$ equal to $0$ .

$4 - x = 0$

Step 4.3.2

Solve $4 - x = 0$ for $x$ .

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Step 4.3.2.1

Subtract $4$ from both sides of the equation.

$- x = - 4$

Step 4.3.2.2

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

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Step 4.3.2.2.1

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

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

Step 4.3.2.2.2

Simplify the left side.

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Step 4.3.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 4.3.2.2.2.2

Divide $x$ by $1$ .

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

Step 4.3.2.2.3

Simplify the right side.

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Step 4.3.2.2.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 4.4

The final solution is all the values that make $(x - 2) (4 - x) = 0$ true.

$x = 2, 4$