Linear Algebra Examples

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

Step 1

Set the denominator in $\frac{1}{x - 1}$ equal to $0$ to find where the expression is undefined.

$x - 1 = 0$

Step 2

Add $1$ to both sides of the equation.

$x = 1$

Step 3

Set the denominator in $\frac{2}{2 x - 1}$ equal to $0$ to find where the expression is undefined.

$2 x - 1 = 0$

Step 4

Solve for $x$ .

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Step 4.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 4.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 4.2.1

Divide each term in $2 x = 1$ by $2$ .

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

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.1.2

Divide $x$ by $1$ .

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

Step 5

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, \frac{1}{2}) \cup (\frac{1}{2}, 1) \cup (1, \infty)$

Set-Builder Notation:

${x | x \neq \frac{1}{2}, 1}$

Step 6