Algebra Examples

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

Step 1

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

$x = 0$

$x - 2 = 0$

Step 2

Add $2$ to both sides of the equation.

$x = 2$

Step 3

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

$x = 0$

$x = 2$

Step 4

Consolidate the solutions.

$x = 0, 2$

Step 5

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

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Step 5.1

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

$x - 2 = 0$

Step 5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 5.3

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

$(- \infty, 2) \cup (2, \infty)$

Step 6

Use each root to create test intervals.

$x < 0$

$0 < x < 2$

$x > 2$

Step 7

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 7.1

Test a value on the interval $x < 0$ to see if it makes the inequality true.

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Step 7.1.1

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 7.1.2

Replace $x$ with $- 2$ in the original inequality.

$\frac{- 2}{(- 2) - 2} > 0$

Step 7.1.3

The left side $0.5$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 7.2

Test a value on the interval $0 < x < 2$ to see if it makes the inequality true.

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Step 7.2.1

Choose a value on the interval $0 < x < 2$ and see if this value makes the original inequality true.

$x = 1$

Step 7.2.2

Replace $x$ with $1$ in the original inequality.

$\frac{1}{(1) - 2} > 0$

Step 7.2.3

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

False

Step 7.3

Test a value on the interval $x > 2$ to see if it makes the inequality true.

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Step 7.3.1

Choose a value on the interval $x > 2$ and see if this value makes the original inequality true.

$x = 4$

Step 7.3.2

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

$\frac{4}{(4) - 2} > 0$

Step 7.3.3

The left side $2$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 7.4

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

$x < 0$ True

$0 < x < 2$ False

$x > 2$ True

$x < 0$ True

$0 < x < 2$ False

$x > 2$ True

Step 8

The solution consists of all of the true intervals.

$x < 0$ or $x > 2$

Step 9

Convert the inequality to interval notation.

$(- \infty, 0) \cup (2, \infty)$

Step 10