Precalculus Examples

$\frac{x - 1}{1 - x} < 0$

Step 1

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$x - 1 = 0$

$1 - x = 0$

Step 2

Add $1$ to both sides of the equation.

$x = 1$

Step 3

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 4

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

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Step 4.1

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

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

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

Step 4.2.2

Divide $x$ by $1$ .

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

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 5

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$x = 1$

Step 6

Consolidate the solutions.

$x = 1, 1$

Step 7

Find the domain of $\frac{x - 1}{1 - x}$ .

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Step 7.1

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

$1 - x = 0$

Step 7.2

Solve for $x$ .

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Step 7.2.1

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 7.2.2

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

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Step 7.2.2.1

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

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

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 7.2.2.2.2

Divide $x$ by $1$ .

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

Step 7.2.2.3

Simplify the right side.

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Step 7.2.2.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 7.3

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

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

Step 8

Use each root to create test intervals.

$x < 1$

$x > 1$

Step 9

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 9.1

Test a value on the interval $x < 1$ to see if it makes the inequality true.

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Step 9.1.1

Choose a value on the interval $x < 1$ and see if this value makes the original inequality true.

$x = 0$

Step 9.1.2

Replace $x$ with $0$ in the original inequality.

$\frac{(0) - 1}{1 - (0)} < 0$

Step 9.1.3

The left side $- 1$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 9.2

Test a value on the interval $x > 1$ to see if it makes the inequality true.

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Step 9.2.1

Choose a value on the interval $x > 1$ and see if this value makes the original inequality true.

$x = 4$

Step 9.2.2

Replace $x$ with $4$ in the original inequality.

$\frac{(4) - 1}{1 - (4)} < 0$

Step 9.2.3

The left side $- 1$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 9.3

Compare the intervals to determine which ones satisfy the original inequality.

$x < 1$ True

$x > 1$ True

$x < 1$ True

$x > 1$ True

Step 10

The solution consists of all of the true intervals.

$x < 1$ or $x > 1$

Step 11

Convert the inequality to interval notation.

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

Step 12