Pre-Algebra Examples

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

Step 1

Subtract $1$ from both sides of the inequality.

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

Step 2

Simplify $\frac{x}{1 - x} - 1$ .

Tap for more steps...

Step 2.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{1 - x}{1 - x}$ .

$\frac{x}{1 - x} - 1 \cdot \frac{1 - x}{1 - x} > 0$

Step 2.2

Combine $- 1$ and $\frac{1 - x}{1 - x}$ .

$\frac{x}{1 - x} + \frac{- (1 - x)}{1 - x} > 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{x - (1 - x)}{1 - x} > 0$

Step 2.4

Simplify the numerator.

Tap for more steps...

Step 2.4.1

Apply the distributive property.

$\frac{x - 1 \cdot 1 - - x}{1 - x} > 0$

Step 2.4.2

Multiply $- 1$ by $1$ .

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

Step 2.4.3

Multiply $- - x$ .

Tap for more steps...

Step 2.4.3.1

Multiply $- 1$ by $- 1$ .

$\frac{x - 1 + 1 x}{1 - x} > 0$

Step 2.4.3.2

Multiply $x$ by $1$ .

$\frac{x - 1 + x}{1 - x} > 0$

Step 2.4.4

Add $x$ and $x$ .

$\frac{2 x - 1}{1 - x} > 0$

Step 3

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

$2 x - 1 = 0$

$1 - x = 0$

Step 4

Add $1$ to both sides of the equation.

$2 x = 1$

Step 5

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

Tap for more steps...

Step 5.1

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

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

Step 5.2

Simplify the left side.

Tap for more steps...

Step 5.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.2.1.1

Cancel the common factor.

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

Step 5.2.1.2

Divide $x$ by $1$ .

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

Step 6

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 7

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

Tap for more steps...

Step 7.1

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

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

Step 7.2

Simplify the left side.

Tap for more steps...

Step 7.2.1

Dividing two negative values results in a positive value.

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

Step 7.2.2

Divide $x$ by $1$ .

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

Step 7.3

Simplify the right side.

Tap for more steps...

Step 7.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 8

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

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

$x = 1$

Step 9

Consolidate the solutions.

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

Step 10

Find the domain of $\frac{2 x - 1}{1 - x}$ .

Tap for more steps...

Step 10.1

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

$1 - x = 0$

Step 10.2

Solve for $x$ .

Tap for more steps...

Step 10.2.1

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 10.2.2

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

Tap for more steps...

Step 10.2.2.1

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

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

Step 10.2.2.2

Simplify the left side.

Tap for more steps...

Step 10.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 10.2.2.2.2

Divide $x$ by $1$ .

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

Step 10.2.2.3

Simplify the right side.

Tap for more steps...

Step 10.2.2.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 10.3

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

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

Step 11

Use each root to create test intervals.

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

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

$x > 1$

Step 12

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

Tap for more steps...

Step 12.1

Test a value on the interval $x < \frac{1}{2}$ to see if it makes the inequality true.

Tap for more steps...

Step 12.1.1

Choose a value on the interval $x < \frac{1}{2}$ and see if this value makes the original inequality true.

$x = 0$

Step 12.1.2

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

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

Step 12.1.3

The left side $0$ is not greater than the right side $1$ , which means that the given statement is false.

False

Step 12.2

Test a value on the interval $\frac{1}{2} < x < 1$ to see if it makes the inequality true.

Tap for more steps...

Step 12.2.1

Choose a value on the interval $\frac{1}{2} < x < 1$ and see if this value makes the original inequality true.

$x = 0.75$

Step 12.2.2

Replace $x$ with $0.75$ in the original inequality.

$\frac{0.75}{1 - (0.75)} > 1$

Step 12.2.3

The left side $3$ is greater than the right side $1$ , which means that the given statement is always true.

True

Step 12.3

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

Tap for more steps...

Step 12.3.1

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

$x = 4$

Step 12.3.2

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

$\frac{4}{1 - (4)} > 1$

Step 12.3.3

The left side $- 1.\overline{3}$ is not greater than the right side $1$ , which means that the given statement is false.

False

Step 12.4

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

$x < \frac{1}{2}$ False

$\frac{1}{2} < x < 1$ True

$x > 1$ False

$x < \frac{1}{2}$ False

$\frac{1}{2} < x < 1$ True

$x > 1$ False

Step 13

The solution consists of all of the true intervals.

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

Step 14

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

$(\frac{1}{2}, 1)$

Step 15