Precalculus Examples

$(m + 1 - \frac{1}{1 - m}) \div (m - \frac{m^{2}}{m - 1})$

Step 1

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

$1 - m = 0$

Step 2

Solve for $m$ .

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Step 2.1

Subtract $1$ from both sides of the equation.

$- m = - 1$

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.2.2.2

Divide $m$ by $1$ .

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 1$ by $- 1$ .

$m = 1$

Step 3

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

$m - 1 = 0$

Step 4

Add $1$ to both sides of the equation.

$m = 1$

Step 5

Set the denominator in $(m + 1 - \frac{1}{1 - m}) \div (m - \frac{m^{2}}{m - 1})$ equal to $0$ to find where the expression is undefined.

$m - \frac{m^{2}}{m - 1} = 0$

Step 6

Solve for $m$ .

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Step 6.1

Find the LCD of the terms in the equation.

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Step 6.1.1

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

$1, m - 1, 1$

Step 6.1.2

Remove parentheses.

$1, m - 1, 1$

Step 6.1.3

The LCM of one and any expression is the expression.

$m - 1$

Step 6.2

Multiply each term in $m - \frac{m^{2}}{m - 1} = 0$ by $m - 1$ to eliminate the fractions.

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Step 6.2.1

Multiply each term in $m - \frac{m^{2}}{m - 1} = 0$ by $m - 1$ .

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

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Simplify each term.

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Step 6.2.2.1.1

Apply the distributive property.

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

Step 6.2.2.1.2

Multiply $m$ by $m$ .

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

Step 6.2.2.1.3

Move $- 1$ to the left of $m$ .

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

Step 6.2.2.1.4

Rewrite $- 1 m$ as $- m$ .

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

Step 6.2.2.1.5

Cancel the common factor of $m - 1$ .

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Step 6.2.2.1.5.1

Move the leading negative in $- \frac{m^{2}}{m - 1}$ into the numerator.

$m^{2} - m + \frac{- m^{2}}{m - 1} (m - 1) = 0 (m - 1)$

Step 6.2.2.1.5.2

Cancel the common factor.

$m^{2} - m + \frac{- m^{2}}{m - 1} (m - 1) = 0 (m - 1)$

Step 6.2.2.1.5.3

Rewrite the expression.

$m^{2} - m - m^{2} = 0 (m - 1)$

Step 6.2.2.2

Combine the opposite terms in $m^{2} - m - m^{2}$ .

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Step 6.2.2.2.1

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

$- m + 0 = 0 (m - 1)$

Step 6.2.2.2.2

Add $- m$ and $0$ .

$- m = 0 (m - 1)$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Multiply $0$ by $m - 1$ .

$- m = 0$

Step 6.3

Divide each term in $- m = 0$ by $- 1$ and simplify.

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Step 6.3.1

Divide each term in $- m = 0$ by $- 1$ .

$\frac{- m}{- 1} = \frac{0}{- 1}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Dividing two negative values results in a positive value.

$\frac{m}{1} = \frac{0}{- 1}$

Step 6.3.2.2

Divide $m$ by $1$ .

$m = \frac{0}{- 1}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Divide $0$ by $- 1$ .

$m = 0$

Step 7

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

Interval Notation:

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

Set-Builder Notation:

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

Step 8