Algebra Examples

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

Step 1

Set $\frac{1}{2 (x - 1)} - 1$ equal to $0$ .

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

Step 2

Solve for $x$ .

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Step 2.1

Add $1$ to both sides of the equation.

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

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

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

$2 (x - 1), 1$

Step 2.2.2

The LCM of one and any expression is the expression.

$2 (x - 1)$

Step 2.3

Multiply each term in $\frac{1}{2 (x - 1)} = 1$ by $2 (x - 1)$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $\frac{1}{2 (x - 1)} = 1$ by $2 (x - 1)$ .

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Rewrite using the commutative property of multiplication.

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

Step 2.3.2.2

Cancel the common factor of $2$ .

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Step 2.3.2.2.1

Cancel the common factor.

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

Step 2.3.2.2.2

Rewrite the expression.

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

Step 2.3.2.3

Cancel the common factor of $x - 1$ .

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Step 2.3.2.3.1

Cancel the common factor.

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

Step 2.3.2.3.2

Rewrite the expression.

$1 = 1 (2 (x - 1))$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Multiply $2 (x - 1)$ by $1$ .

$1 = 2 (x - 1)$

Step 2.3.3.2

Apply the distributive property.

$1 = 2 x + 2 \cdot - 1$

Step 2.3.3.3

Multiply $2$ by $- 1$ .

$1 = 2 x - 2$

Step 2.4

Solve the equation.

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Step 2.4.1

Rewrite the equation as $2 x - 2 = 1$ .

$2 x - 2 = 1$

Step 2.4.2

Move all terms not containing $x$ to the right side of the equation.

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Step 2.4.2.1

Add $2$ to both sides of the equation.

$2 x = 1 + 2$

Step 2.4.2.2

Add $1$ and $2$ .

$2 x = 3$

Step 2.4.3

Divide each term in $2 x = 3$ by $2$ and simplify.

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Step 2.4.3.1

Divide each term in $2 x = 3$ by $2$ .

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

Step 2.4.3.2

Simplify the left side.

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Step 2.4.3.2.1

Cancel the common factor of $2$ .

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Step 2.4.3.2.1.1

Cancel the common factor.

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

Step 2.4.3.2.1.2

Divide $x$ by $1$ .

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

Step 3