Precalculus Examples

$\frac{a - \frac{2 a - 1}{a}}{\frac{1 - a}{3 a}}$

Step 1

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

$a = 0$

Step 2

Set the denominator in $\frac{1 - a}{3 a}$ equal to $0$ to find where the expression is undefined.

$3 a = 0$

Step 3

Divide each term in $3 a = 0$ by $3$ and simplify.

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Step 3.1

Divide each term in $3 a = 0$ by $3$ .

$\frac{3 a}{3} = \frac{0}{3}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $3$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{3 a}{3} = \frac{0}{3}$

Step 3.2.1.2

Divide $a$ by $1$ .

$a = \frac{0}{3}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $3$ .

$a = 0$

Step 4

Set the denominator in $\frac{a - \frac{2 a - 1}{a}}{\frac{1 - a}{3 a}}$ equal to $0$ to find where the expression is undefined.

$\frac{1 - a}{3 a} = 0$

Step 5

Solve for $a$ .

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Step 5.1

Set the numerator equal to zero.

$1 - a = 0$

Step 5.2

Solve the equation for $a$ .

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Step 5.2.1

Subtract $1$ from both sides of the equation.

$- a = - 1$

Step 5.2.2

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

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Step 5.2.2.1

Divide each term in $- a = - 1$ by $- 1$ .

$\frac{- a}{- 1} = \frac{- 1}{- 1}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{a}{1} = \frac{- 1}{- 1}$

Step 5.2.2.2.2

Divide $a$ by $1$ .

$a = \frac{- 1}{- 1}$

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Divide $- 1$ by $- 1$ .

$a = 1$

Step 6

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

Interval Notation:

$(- \infty, 0) \cup (0, 1) \cup (1, \infty)$

Set-Builder Notation:

${a | a \neq 0, 1}$

Step 7