Convert to Interval Notation x/(x-10)<0

$\frac{x}{x - 10} < 0$

Step 1

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

$x = 0$

$x - 10 = 0$

Step 2

Add $10$ to both sides of the equation.

$x = 10$

Step 3

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

$x = 0$

$x = 10$

Step 4

Consolidate the solutions.

$x = 0, 10$

Step 5

Find the domain of $\frac{x}{x - 10}$ .

Set the denominator in $\frac{x}{x - 10}$ equal to $0$ to find where the expression is undefined.

$x - 10 = 0$

Add $10$ to both sides of the equation.

$x = 10$

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

$(- \infty, 10) \cup (10, \infty)$

Step 6

Use each root to create test intervals.

$x < 0$

$0 < x < 10$

$x > 10$

Step 7

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

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

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

$x = - 2$

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

$\frac{- 2}{(- 2) - 10} < 0$

The left side $0.1\overline{6}$ is not less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $0 < x < 10$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < 10$ and see if this value makes the original inequality true.

$x = 5$

Replace $x$ with $5$ in the original inequality.

$\frac{5}{(5) - 10} < 0$

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

True

Test a value on the interval $x > 10$ to see if it makes the inequality true.

Choose a value on the interval $x > 10$ and see if this value makes the original inequality true.

$x = 12$

Replace $x$ with $12$ in the original inequality.

$\frac{12}{(12) - 10} < 0$

The left side $6$ is not less than the right side $0$ , which means that the given statement is false.

False

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

$x < 0$ False

$0 < x < 10$ True

$x > 10$ False

$x < 0$ False

$0 < x < 10$ True

$x > 10$ False

Step 8

The solution consists of all of the true intervals.

$0 < x < 10$

Step 9

Convert the inequality to interval notation.

$(0, 10)$

Step 10

Convert to Interval Notation x/(x-10)<0

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