# Calculus Examples

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

Add to both sides of the equation.

Subtract from both sides of the equation.

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

Consolidate the solutions.

Solve to find the value of that makes the expression undefined.

Set up the equation to solve for .

Subtract from both sides of the equation.

For asymptotes and point discontinuity (denominator equals ), it is easier to find where the expression is undefined. These values are not part of the domain.

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

Use each root to create test intervals.

Test a value on the interval to see if it makes the inequality true.

Choose a value on the interval and see if this value makes the original inequality true.

Replace with in the original inequality.

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

True

True

Test a value on the interval to see if it makes the inequality true.

Choose a value on the interval and see if this value makes the original inequality true.

Replace with in the original inequality.

The left side is less than the right side , which means that the given statement is false.

False

False

Test a value on the interval to see if it makes the inequality true.

Choose a value on the interval and see if this value makes the original inequality true.

Replace with in the original inequality.

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

True

True

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

True

False

True

True

False

True

The solution consists of all of the true intervals.

Remove any values from the solution that make the denominator equal to .

or